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|---|---|---|---|---|---|---|
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : multiplicity p a ≤ multiplicity p b
hab : Coprime a b
⊢ multiplicity p a = 0
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
/-!
# Multiplicity of a divisor
For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves
several basic results on it.
## Main definitions
* `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest
number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers
`n`.
* `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite.
-/
variable {α β : Type*}
open Nat Part
open BigOperators
/-- `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤`-/
def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat :=
PartENat.find fun n => ¬a ^ (n + 1) ∣ b
#align multiplicity multiplicity
namespace multiplicity
section Monoid
variable [Monoid α] [Monoid β]
/-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/
@[reducible]
def Finite (a b : α) : Prop :=
∃ n : ℕ, ¬a ^ (n + 1) ∣ b
#align multiplicity.finite multiplicity.Finite
theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} :
Finite a b ↔ (multiplicity a b).Dom :=
Iff.rfl
#align multiplicity.finite_iff_dom multiplicity.finite_iff_dom
theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b :=
Iff.rfl
#align multiplicity.finite_def multiplicity.finite_def
theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ =>
hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩
#align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right
@[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
simp
#align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity
theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by simpa [Finite, Classical.not_not] using h),
by simp [Finite, multiplicity, Classical.not_not]; tauto⟩
#align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall
theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a :=
let ⟨n, hn⟩ := h
hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1)
#align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite
theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ =>
⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩
#align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right
variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)]
theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
Nat.casesOn k
(fun _ => by
rw [_root_.pow_zero]
exact one_dvd _)
fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk
#align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity
theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get])
#align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd
theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by
rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h)
#align multiplicity.is_greatest multiplicity.is_greatest
theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm)
#align multiplicity.is_greatest' multiplicity.is_greatest'
theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by
refine' zero_lt_iff.2 fun h => _
simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h)
#align multiplicity.pos_of_dvd multiplicity.pos_of_dvd
theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
have : Finite a b := ⟨k, hsucc⟩
rw [PartENat.le_coe_iff]
exact ⟨this, Nat.find_min' _ hsucc⟩
#align multiplicity.unique multiplicity.unique
theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ := by
rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc]
#align multiplicity.unique' multiplicity.unique'
theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) :
(k : PartENat) ≤ multiplicity a b :=
le_of_not_gt fun hk' => is_greatest hk' hk
#align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd
theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} :
a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b :=
⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩
#align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity
theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le]
#align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd
theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
rw [PartENat.lt_coe_iff]
exact ⟨h₁, lt_succ_self _⟩)⟩,
fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩
#align multiplicity.eq_coe_iff multiplicity.eq_coe_iff
theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
exact
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
fun n => h _,
fun h n => h _⟩
#align multiplicity.eq_top_iff multiplicity.eq_top_iff
@[simp]
theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ :=
eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _)
#align multiplicity.is_unit_left multiplicity.isUnit_left
-- @[simp] Porting note: simp can prove this
theorem one_left (b : α) : multiplicity 1 b = ⊤ :=
isUnit_left b isUnit_one
#align multiplicity.one_left multiplicity.one_left
@[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by
rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero]
simp [not_dvd_one_of_finite_one_right ha]
#align multiplicity.get_one_right multiplicity.get_one_right
-- @[simp] Porting note: simp can prove this
theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ :=
isUnit_left a u.isUnit
#align multiplicity.unit_left multiplicity.unit_left
theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by
rw [← Nat.cast_zero, eq_coe_iff]
simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and]
#align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero
theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b :=
multiplicity_eq_zero.not_left
#align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero
theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b :=
Part.eq_none_iff'
#align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite
theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by
rw [Ne.def, eq_top_iff_not_finite, Classical.not_not]
#align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite
theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by
rw [lt_top_iff_ne_top, ne_top_iff_finite]
#align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite
theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩
exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁
#align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd
theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else by
have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _)
rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩
#align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff
theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d :=
⟨fun h n =>
⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩,
fun h =>
le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp)
(multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩
#align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff
theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) :=
multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f
theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b :=
multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f
theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right
theorem eq_of_associated_right {a b c : α} (h : Associated b c) :
multiplicity a b = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd)
(multiplicity_le_multiplicity_of_dvd_right h.symm.dvd)
#align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right
theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
rw [← pow_one a]
apply pow_dvd_of_le_multiplicity
simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h
#align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos
theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos, fun hdvd =>
lt_of_le_of_ne (zero_le _) fun heq =>
is_greatest
(show multiplicity a b < ↑1 by
simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one)
(by rwa [pow_one a])⟩
#align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos
theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1
Classical.by_contradiction fun ha1 : a ≠ 1 =>
have ha_gt_one : 1 < a :=
lt_of_not_ge fun _ =>
match a with
| 0 => ha rfl
| 1 => ha1 rfl
| b+2 => by linarith
not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b),
fun h => by cases h <;> simp [*]⟩
#align multiplicity.finite_nat_iff multiplicity.finite_nat_iff
alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos
end Monoid
section CommMonoid
variable [CommMonoid α]
theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by
rw [mul_comm]; exact finite_of_finite_mul_right
#align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by
rw [pow_one]
exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩
#align multiplicity.is_unit_right multiplicity.isUnit_right
theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 :=
isUnit_right ha isUnit_one
#align multiplicity.one_right multiplicity.one_right
theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 :=
isUnit_right ha u.isUnit
#align multiplicity.unit_right multiplicity.unit_right
open Classical
theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left
theorem eq_of_associated_left {a b c : α} (h : Associated a b) :
multiplicity b c = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd)
(multiplicity_le_multiplicity_of_dvd_left h.symm.dvd)
#align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left
-- Porting note: this was doing nothing in mathlib3 also
-- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos
end CommMonoid
section MonoidWithZero
variable [MonoidWithZero α]
theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h
fun hb => by simp [hb] at hn
#align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem zero (a : α) : multiplicity a 0 = ⊤ :=
Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _)
#align multiplicity.zero multiplicity.zero
@[simp]
theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 :=
multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha
#align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero
end MonoidWithZero
section CommMonoidWithZero
variable [CommMonoidWithZero α]
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
(show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;>
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
· exact pow_multiplicity_dvd h
· exact is_greatest
((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
· suffices ¬Finite (Associates.mk a) (Associates.mk b) by
rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
rw [h, this]
refine'
not_finite_iff_forall.mpr fun n => by
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
exact not_finite_iff_forall.mp h n
#align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity
end CommMonoidWithZero
section Semiring
variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
fun h => by
rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn
#align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add
end Semiring
section Ring
variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ =>
PartENat.natCast_inj.1 (by
rw [PartENat.natCast_get]
exact Eq.symm
(unique (pow_multiplicity_dvd _).neg_right
(mt dvd_neg.1 (is_greatest' _ (lt_succ_self _)))))
#align multiplicity.neg multiplicity.neg
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h]
· rw [Int.coe_nat_multiplicity]
· rw [multiplicity.neg, Int.coe_nat_multiplicity]
#align multiplicity.int.nat_abs multiplicity.Int.natAbs
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
intro h_dvd
· apply multiplicity.is_greatest _ h_dvd
rw [hk, ← Nat.succ_eq_add_one]
norm_cast
apply Nat.lt_succ_self k
· rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
exact PartENat.add_one_le_of_lt h
· have := @min_le_multiplicity_add α _ _ p a b
rwa [← min_eq_right (le_of_lt h)]
#align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption
#align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
· contradiction
· rw [multiplicity_add_of_gt hab, min_eq_right]
exact le_of_lt hab
#align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min
end Ring
section CancelCommMonoidWithZero
variable [CancelCommMonoidWithZero α]
/- Porting note: removed previous wf recursion hints and added termination_by
Also pulled a b intro parameters since Lean parses that more easily -/
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>
have hn0 : 0 < n :=
Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
finite_mul_aux hp hpx hb
⟨s, mul_right_cancel₀ hp.1 (by
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
fun ⟨x, hx⟩ =>
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
hb
(hx.symm ▸
⟨y,
mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
finite_mul_aux hp ha hpx
⟨s, mul_right_cancel₀ hp.1 (by
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩
termination_by finite_mul_aux _ _ n m => n+m
#align multiplicity.finite_mul_aux multiplicity.finite_mul_aux
theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) :=
fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩
#align multiplicity.finite_mul multiplicity.finite_mul
theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b :=
⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h =>
finite_mul hp h.1 h.2⟩
#align multiplicity.finite_mul_iff multiplicity.finite_mul_iff
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
#align multiplicity.finite_pow multiplicity.finite_pow
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩
#align multiplicity.multiplicity_self multiplicity.multiplicity_self
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact
mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩)
#align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
have hpoweq :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
by simp [pow_add]
have hdiv :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
a * b :=
by rw [hpoweq]; apply mul_dvd_mul <;> assumption
have hsucc :
¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 +
1) ∣
a * b :=
fun h =>
not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩
#align multiplicity.mul' multiplicity.mul'
open Classical
protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add,
PartENat.natCast_inj, multiplicity.mul' hp]
else by
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h]
#align multiplicity.mul multiplicity.mul
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
· simp [has, ← ih]
convert multiplicity.mul hp
#align multiplicity.finset.prod multiplicity.Finset.prod
-- Porting note: with protected could not use pow' k in the succ branch
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
rw [succ_eq_add_one, get_eq_get_of_eq _ _ this,
multiplicity.mul' hp, hk, add_mul, one_mul, add_comm]
#align multiplicity.pow' multiplicity.pow'
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
#align multiplicity.pow multiplicity.pow
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
apply Nat.not_succ_le_self
#align multiplicity.multiplicity_pow_self multiplicity.multiplicity_pow_self
theorem multiplicity_pow_self_of_prime {p : α} (hp : Prime p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
multiplicity_pow_self hp.ne_zero hp.not_unit n
#align multiplicity.multiplicity_pow_self_of_prime multiplicity.multiplicity_pow_self_of_prime
end CancelCommMonoidWithZero
end multiplicity
section Nat
open multiplicity
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
|
rw [multiplicity_le_multiplicity_iff] at hle
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
|
Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0
|
Mathlib_RingTheory_Multiplicity
|
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
⊢ multiplicity p a = 0
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
/-!
# Multiplicity of a divisor
For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves
several basic results on it.
## Main definitions
* `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest
number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers
`n`.
* `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite.
-/
variable {α β : Type*}
open Nat Part
open BigOperators
/-- `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤`-/
def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat :=
PartENat.find fun n => ¬a ^ (n + 1) ∣ b
#align multiplicity multiplicity
namespace multiplicity
section Monoid
variable [Monoid α] [Monoid β]
/-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/
@[reducible]
def Finite (a b : α) : Prop :=
∃ n : ℕ, ¬a ^ (n + 1) ∣ b
#align multiplicity.finite multiplicity.Finite
theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} :
Finite a b ↔ (multiplicity a b).Dom :=
Iff.rfl
#align multiplicity.finite_iff_dom multiplicity.finite_iff_dom
theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b :=
Iff.rfl
#align multiplicity.finite_def multiplicity.finite_def
theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ =>
hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩
#align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right
@[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
simp
#align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity
theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by simpa [Finite, Classical.not_not] using h),
by simp [Finite, multiplicity, Classical.not_not]; tauto⟩
#align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall
theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a :=
let ⟨n, hn⟩ := h
hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1)
#align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite
theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ =>
⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩
#align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right
variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)]
theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
Nat.casesOn k
(fun _ => by
rw [_root_.pow_zero]
exact one_dvd _)
fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk
#align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity
theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get])
#align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd
theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by
rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h)
#align multiplicity.is_greatest multiplicity.is_greatest
theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm)
#align multiplicity.is_greatest' multiplicity.is_greatest'
theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by
refine' zero_lt_iff.2 fun h => _
simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h)
#align multiplicity.pos_of_dvd multiplicity.pos_of_dvd
theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
have : Finite a b := ⟨k, hsucc⟩
rw [PartENat.le_coe_iff]
exact ⟨this, Nat.find_min' _ hsucc⟩
#align multiplicity.unique multiplicity.unique
theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ := by
rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc]
#align multiplicity.unique' multiplicity.unique'
theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) :
(k : PartENat) ≤ multiplicity a b :=
le_of_not_gt fun hk' => is_greatest hk' hk
#align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd
theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} :
a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b :=
⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩
#align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity
theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le]
#align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd
theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
rw [PartENat.lt_coe_iff]
exact ⟨h₁, lt_succ_self _⟩)⟩,
fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩
#align multiplicity.eq_coe_iff multiplicity.eq_coe_iff
theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
exact
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
fun n => h _,
fun h n => h _⟩
#align multiplicity.eq_top_iff multiplicity.eq_top_iff
@[simp]
theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ :=
eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _)
#align multiplicity.is_unit_left multiplicity.isUnit_left
-- @[simp] Porting note: simp can prove this
theorem one_left (b : α) : multiplicity 1 b = ⊤ :=
isUnit_left b isUnit_one
#align multiplicity.one_left multiplicity.one_left
@[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by
rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero]
simp [not_dvd_one_of_finite_one_right ha]
#align multiplicity.get_one_right multiplicity.get_one_right
-- @[simp] Porting note: simp can prove this
theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ :=
isUnit_left a u.isUnit
#align multiplicity.unit_left multiplicity.unit_left
theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by
rw [← Nat.cast_zero, eq_coe_iff]
simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and]
#align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero
theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b :=
multiplicity_eq_zero.not_left
#align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero
theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b :=
Part.eq_none_iff'
#align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite
theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by
rw [Ne.def, eq_top_iff_not_finite, Classical.not_not]
#align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite
theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by
rw [lt_top_iff_ne_top, ne_top_iff_finite]
#align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite
theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩
exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁
#align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd
theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else by
have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _)
rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩
#align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff
theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d :=
⟨fun h n =>
⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩,
fun h =>
le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp)
(multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩
#align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff
theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) :=
multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f
theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b :=
multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f
theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right
theorem eq_of_associated_right {a b c : α} (h : Associated b c) :
multiplicity a b = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd)
(multiplicity_le_multiplicity_of_dvd_right h.symm.dvd)
#align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right
theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
rw [← pow_one a]
apply pow_dvd_of_le_multiplicity
simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h
#align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos
theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos, fun hdvd =>
lt_of_le_of_ne (zero_le _) fun heq =>
is_greatest
(show multiplicity a b < ↑1 by
simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one)
(by rwa [pow_one a])⟩
#align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos
theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1
Classical.by_contradiction fun ha1 : a ≠ 1 =>
have ha_gt_one : 1 < a :=
lt_of_not_ge fun _ =>
match a with
| 0 => ha rfl
| 1 => ha1 rfl
| b+2 => by linarith
not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b),
fun h => by cases h <;> simp [*]⟩
#align multiplicity.finite_nat_iff multiplicity.finite_nat_iff
alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos
end Monoid
section CommMonoid
variable [CommMonoid α]
theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by
rw [mul_comm]; exact finite_of_finite_mul_right
#align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by
rw [pow_one]
exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩
#align multiplicity.is_unit_right multiplicity.isUnit_right
theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 :=
isUnit_right ha isUnit_one
#align multiplicity.one_right multiplicity.one_right
theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 :=
isUnit_right ha u.isUnit
#align multiplicity.unit_right multiplicity.unit_right
open Classical
theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left
theorem eq_of_associated_left {a b c : α} (h : Associated a b) :
multiplicity b c = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd)
(multiplicity_le_multiplicity_of_dvd_left h.symm.dvd)
#align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left
-- Porting note: this was doing nothing in mathlib3 also
-- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos
end CommMonoid
section MonoidWithZero
variable [MonoidWithZero α]
theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h
fun hb => by simp [hb] at hn
#align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem zero (a : α) : multiplicity a 0 = ⊤ :=
Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _)
#align multiplicity.zero multiplicity.zero
@[simp]
theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 :=
multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha
#align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero
end MonoidWithZero
section CommMonoidWithZero
variable [CommMonoidWithZero α]
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
(show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;>
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
· exact pow_multiplicity_dvd h
· exact is_greatest
((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
· suffices ¬Finite (Associates.mk a) (Associates.mk b) by
rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
rw [h, this]
refine'
not_finite_iff_forall.mpr fun n => by
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
exact not_finite_iff_forall.mp h n
#align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity
end CommMonoidWithZero
section Semiring
variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
fun h => by
rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn
#align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add
end Semiring
section Ring
variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ =>
PartENat.natCast_inj.1 (by
rw [PartENat.natCast_get]
exact Eq.symm
(unique (pow_multiplicity_dvd _).neg_right
(mt dvd_neg.1 (is_greatest' _ (lt_succ_self _)))))
#align multiplicity.neg multiplicity.neg
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h]
· rw [Int.coe_nat_multiplicity]
· rw [multiplicity.neg, Int.coe_nat_multiplicity]
#align multiplicity.int.nat_abs multiplicity.Int.natAbs
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
intro h_dvd
· apply multiplicity.is_greatest _ h_dvd
rw [hk, ← Nat.succ_eq_add_one]
norm_cast
apply Nat.lt_succ_self k
· rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
exact PartENat.add_one_le_of_lt h
· have := @min_le_multiplicity_add α _ _ p a b
rwa [← min_eq_right (le_of_lt h)]
#align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption
#align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
· contradiction
· rw [multiplicity_add_of_gt hab, min_eq_right]
exact le_of_lt hab
#align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min
end Ring
section CancelCommMonoidWithZero
variable [CancelCommMonoidWithZero α]
/- Porting note: removed previous wf recursion hints and added termination_by
Also pulled a b intro parameters since Lean parses that more easily -/
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>
have hn0 : 0 < n :=
Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
finite_mul_aux hp hpx hb
⟨s, mul_right_cancel₀ hp.1 (by
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
fun ⟨x, hx⟩ =>
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
hb
(hx.symm ▸
⟨y,
mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
finite_mul_aux hp ha hpx
⟨s, mul_right_cancel₀ hp.1 (by
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩
termination_by finite_mul_aux _ _ n m => n+m
#align multiplicity.finite_mul_aux multiplicity.finite_mul_aux
theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) :=
fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩
#align multiplicity.finite_mul multiplicity.finite_mul
theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b :=
⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h =>
finite_mul hp h.1 h.2⟩
#align multiplicity.finite_mul_iff multiplicity.finite_mul_iff
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
#align multiplicity.finite_pow multiplicity.finite_pow
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩
#align multiplicity.multiplicity_self multiplicity.multiplicity_self
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact
mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩)
#align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
have hpoweq :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
by simp [pow_add]
have hdiv :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
a * b :=
by rw [hpoweq]; apply mul_dvd_mul <;> assumption
have hsucc :
¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 +
1) ∣
a * b :=
fun h =>
not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩
#align multiplicity.mul' multiplicity.mul'
open Classical
protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add,
PartENat.natCast_inj, multiplicity.mul' hp]
else by
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h]
#align multiplicity.mul multiplicity.mul
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
· simp [has, ← ih]
convert multiplicity.mul hp
#align multiplicity.finset.prod multiplicity.Finset.prod
-- Porting note: with protected could not use pow' k in the succ branch
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
rw [succ_eq_add_one, get_eq_get_of_eq _ _ this,
multiplicity.mul' hp, hk, add_mul, one_mul, add_comm]
#align multiplicity.pow' multiplicity.pow'
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
#align multiplicity.pow multiplicity.pow
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
apply Nat.not_succ_le_self
#align multiplicity.multiplicity_pow_self multiplicity.multiplicity_pow_self
theorem multiplicity_pow_self_of_prime {p : α} (hp : Prime p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
multiplicity_pow_self hp.ne_zero hp.not_unit n
#align multiplicity.multiplicity_pow_self_of_prime multiplicity.multiplicity_pow_self_of_prime
end CancelCommMonoidWithZero
end multiplicity
section Nat
open multiplicity
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
|
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
|
Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0
|
Mathlib_RingTheory_Multiplicity
|
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
⊢ ¬p ^ 1 ∣ a
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
/-!
# Multiplicity of a divisor
For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves
several basic results on it.
## Main definitions
* `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest
number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers
`n`.
* `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite.
-/
variable {α β : Type*}
open Nat Part
open BigOperators
/-- `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤`-/
def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat :=
PartENat.find fun n => ¬a ^ (n + 1) ∣ b
#align multiplicity multiplicity
namespace multiplicity
section Monoid
variable [Monoid α] [Monoid β]
/-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/
@[reducible]
def Finite (a b : α) : Prop :=
∃ n : ℕ, ¬a ^ (n + 1) ∣ b
#align multiplicity.finite multiplicity.Finite
theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} :
Finite a b ↔ (multiplicity a b).Dom :=
Iff.rfl
#align multiplicity.finite_iff_dom multiplicity.finite_iff_dom
theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b :=
Iff.rfl
#align multiplicity.finite_def multiplicity.finite_def
theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ =>
hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩
#align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right
@[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
simp
#align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity
theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by simpa [Finite, Classical.not_not] using h),
by simp [Finite, multiplicity, Classical.not_not]; tauto⟩
#align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall
theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a :=
let ⟨n, hn⟩ := h
hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1)
#align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite
theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ =>
⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩
#align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right
variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)]
theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
Nat.casesOn k
(fun _ => by
rw [_root_.pow_zero]
exact one_dvd _)
fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk
#align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity
theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get])
#align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd
theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by
rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h)
#align multiplicity.is_greatest multiplicity.is_greatest
theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm)
#align multiplicity.is_greatest' multiplicity.is_greatest'
theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by
refine' zero_lt_iff.2 fun h => _
simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h)
#align multiplicity.pos_of_dvd multiplicity.pos_of_dvd
theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
have : Finite a b := ⟨k, hsucc⟩
rw [PartENat.le_coe_iff]
exact ⟨this, Nat.find_min' _ hsucc⟩
#align multiplicity.unique multiplicity.unique
theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ := by
rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc]
#align multiplicity.unique' multiplicity.unique'
theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) :
(k : PartENat) ≤ multiplicity a b :=
le_of_not_gt fun hk' => is_greatest hk' hk
#align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd
theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} :
a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b :=
⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩
#align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity
theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le]
#align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd
theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
rw [PartENat.lt_coe_iff]
exact ⟨h₁, lt_succ_self _⟩)⟩,
fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩
#align multiplicity.eq_coe_iff multiplicity.eq_coe_iff
theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
exact
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
fun n => h _,
fun h n => h _⟩
#align multiplicity.eq_top_iff multiplicity.eq_top_iff
@[simp]
theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ :=
eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _)
#align multiplicity.is_unit_left multiplicity.isUnit_left
-- @[simp] Porting note: simp can prove this
theorem one_left (b : α) : multiplicity 1 b = ⊤ :=
isUnit_left b isUnit_one
#align multiplicity.one_left multiplicity.one_left
@[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by
rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero]
simp [not_dvd_one_of_finite_one_right ha]
#align multiplicity.get_one_right multiplicity.get_one_right
-- @[simp] Porting note: simp can prove this
theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ :=
isUnit_left a u.isUnit
#align multiplicity.unit_left multiplicity.unit_left
theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by
rw [← Nat.cast_zero, eq_coe_iff]
simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and]
#align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero
theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b :=
multiplicity_eq_zero.not_left
#align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero
theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b :=
Part.eq_none_iff'
#align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite
theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by
rw [Ne.def, eq_top_iff_not_finite, Classical.not_not]
#align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite
theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by
rw [lt_top_iff_ne_top, ne_top_iff_finite]
#align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite
theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩
exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁
#align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd
theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else by
have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _)
rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩
#align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff
theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d :=
⟨fun h n =>
⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩,
fun h =>
le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp)
(multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩
#align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff
theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) :=
multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f
theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b :=
multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f
theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right
theorem eq_of_associated_right {a b c : α} (h : Associated b c) :
multiplicity a b = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd)
(multiplicity_le_multiplicity_of_dvd_right h.symm.dvd)
#align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right
theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
rw [← pow_one a]
apply pow_dvd_of_le_multiplicity
simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h
#align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos
theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos, fun hdvd =>
lt_of_le_of_ne (zero_le _) fun heq =>
is_greatest
(show multiplicity a b < ↑1 by
simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one)
(by rwa [pow_one a])⟩
#align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos
theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1
Classical.by_contradiction fun ha1 : a ≠ 1 =>
have ha_gt_one : 1 < a :=
lt_of_not_ge fun _ =>
match a with
| 0 => ha rfl
| 1 => ha1 rfl
| b+2 => by linarith
not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b),
fun h => by cases h <;> simp [*]⟩
#align multiplicity.finite_nat_iff multiplicity.finite_nat_iff
alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos
end Monoid
section CommMonoid
variable [CommMonoid α]
theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by
rw [mul_comm]; exact finite_of_finite_mul_right
#align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by
rw [pow_one]
exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩
#align multiplicity.is_unit_right multiplicity.isUnit_right
theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 :=
isUnit_right ha isUnit_one
#align multiplicity.one_right multiplicity.one_right
theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 :=
isUnit_right ha u.isUnit
#align multiplicity.unit_right multiplicity.unit_right
open Classical
theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left
theorem eq_of_associated_left {a b c : α} (h : Associated a b) :
multiplicity b c = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd)
(multiplicity_le_multiplicity_of_dvd_left h.symm.dvd)
#align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left
-- Porting note: this was doing nothing in mathlib3 also
-- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos
end CommMonoid
section MonoidWithZero
variable [MonoidWithZero α]
theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h
fun hb => by simp [hb] at hn
#align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem zero (a : α) : multiplicity a 0 = ⊤ :=
Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _)
#align multiplicity.zero multiplicity.zero
@[simp]
theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 :=
multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha
#align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero
end MonoidWithZero
section CommMonoidWithZero
variable [CommMonoidWithZero α]
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
(show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;>
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
· exact pow_multiplicity_dvd h
· exact is_greatest
((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
· suffices ¬Finite (Associates.mk a) (Associates.mk b) by
rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
rw [h, this]
refine'
not_finite_iff_forall.mpr fun n => by
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
exact not_finite_iff_forall.mp h n
#align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity
end CommMonoidWithZero
section Semiring
variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
fun h => by
rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn
#align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add
end Semiring
section Ring
variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ =>
PartENat.natCast_inj.1 (by
rw [PartENat.natCast_get]
exact Eq.symm
(unique (pow_multiplicity_dvd _).neg_right
(mt dvd_neg.1 (is_greatest' _ (lt_succ_self _)))))
#align multiplicity.neg multiplicity.neg
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h]
· rw [Int.coe_nat_multiplicity]
· rw [multiplicity.neg, Int.coe_nat_multiplicity]
#align multiplicity.int.nat_abs multiplicity.Int.natAbs
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
intro h_dvd
· apply multiplicity.is_greatest _ h_dvd
rw [hk, ← Nat.succ_eq_add_one]
norm_cast
apply Nat.lt_succ_self k
· rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
exact PartENat.add_one_le_of_lt h
· have := @min_le_multiplicity_add α _ _ p a b
rwa [← min_eq_right (le_of_lt h)]
#align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption
#align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
· contradiction
· rw [multiplicity_add_of_gt hab, min_eq_right]
exact le_of_lt hab
#align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min
end Ring
section CancelCommMonoidWithZero
variable [CancelCommMonoidWithZero α]
/- Porting note: removed previous wf recursion hints and added termination_by
Also pulled a b intro parameters since Lean parses that more easily -/
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>
have hn0 : 0 < n :=
Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
finite_mul_aux hp hpx hb
⟨s, mul_right_cancel₀ hp.1 (by
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
fun ⟨x, hx⟩ =>
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
hb
(hx.symm ▸
⟨y,
mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
finite_mul_aux hp ha hpx
⟨s, mul_right_cancel₀ hp.1 (by
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩
termination_by finite_mul_aux _ _ n m => n+m
#align multiplicity.finite_mul_aux multiplicity.finite_mul_aux
theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) :=
fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩
#align multiplicity.finite_mul multiplicity.finite_mul
theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b :=
⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h =>
finite_mul hp h.1 h.2⟩
#align multiplicity.finite_mul_iff multiplicity.finite_mul_iff
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
#align multiplicity.finite_pow multiplicity.finite_pow
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩
#align multiplicity.multiplicity_self multiplicity.multiplicity_self
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact
mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩)
#align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
have hpoweq :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
by simp [pow_add]
have hdiv :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
a * b :=
by rw [hpoweq]; apply mul_dvd_mul <;> assumption
have hsucc :
¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 +
1) ∣
a * b :=
fun h =>
not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩
#align multiplicity.mul' multiplicity.mul'
open Classical
protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add,
PartENat.natCast_inj, multiplicity.mul' hp]
else by
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h]
#align multiplicity.mul multiplicity.mul
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
· simp [has, ← ih]
convert multiplicity.mul hp
#align multiplicity.finset.prod multiplicity.Finset.prod
-- Porting note: with protected could not use pow' k in the succ branch
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
rw [succ_eq_add_one, get_eq_get_of_eq _ _ this,
multiplicity.mul' hp, hk, add_mul, one_mul, add_comm]
#align multiplicity.pow' multiplicity.pow'
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
#align multiplicity.pow multiplicity.pow
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
apply Nat.not_succ_le_self
#align multiplicity.multiplicity_pow_self multiplicity.multiplicity_pow_self
theorem multiplicity_pow_self_of_prime {p : α} (hp : Prime p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
multiplicity_pow_self hp.ne_zero hp.not_unit n
#align multiplicity.multiplicity_pow_self_of_prime multiplicity.multiplicity_pow_self_of_prime
end CancelCommMonoidWithZero
end multiplicity
section Nat
open multiplicity
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
|
intro h
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
|
Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0
|
Mathlib_RingTheory_Multiplicity
|
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
h : p ^ 1 ∣ a
⊢ False
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
/-!
# Multiplicity of a divisor
For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves
several basic results on it.
## Main definitions
* `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest
number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers
`n`.
* `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite.
-/
variable {α β : Type*}
open Nat Part
open BigOperators
/-- `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤`-/
def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat :=
PartENat.find fun n => ¬a ^ (n + 1) ∣ b
#align multiplicity multiplicity
namespace multiplicity
section Monoid
variable [Monoid α] [Monoid β]
/-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/
@[reducible]
def Finite (a b : α) : Prop :=
∃ n : ℕ, ¬a ^ (n + 1) ∣ b
#align multiplicity.finite multiplicity.Finite
theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} :
Finite a b ↔ (multiplicity a b).Dom :=
Iff.rfl
#align multiplicity.finite_iff_dom multiplicity.finite_iff_dom
theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b :=
Iff.rfl
#align multiplicity.finite_def multiplicity.finite_def
theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ =>
hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩
#align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right
@[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
simp
#align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity
theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by simpa [Finite, Classical.not_not] using h),
by simp [Finite, multiplicity, Classical.not_not]; tauto⟩
#align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall
theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a :=
let ⟨n, hn⟩ := h
hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1)
#align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite
theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ =>
⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩
#align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right
variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)]
theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
Nat.casesOn k
(fun _ => by
rw [_root_.pow_zero]
exact one_dvd _)
fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk
#align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity
theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get])
#align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd
theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by
rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h)
#align multiplicity.is_greatest multiplicity.is_greatest
theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm)
#align multiplicity.is_greatest' multiplicity.is_greatest'
theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by
refine' zero_lt_iff.2 fun h => _
simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h)
#align multiplicity.pos_of_dvd multiplicity.pos_of_dvd
theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
have : Finite a b := ⟨k, hsucc⟩
rw [PartENat.le_coe_iff]
exact ⟨this, Nat.find_min' _ hsucc⟩
#align multiplicity.unique multiplicity.unique
theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ := by
rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc]
#align multiplicity.unique' multiplicity.unique'
theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) :
(k : PartENat) ≤ multiplicity a b :=
le_of_not_gt fun hk' => is_greatest hk' hk
#align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd
theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} :
a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b :=
⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩
#align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity
theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le]
#align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd
theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
rw [PartENat.lt_coe_iff]
exact ⟨h₁, lt_succ_self _⟩)⟩,
fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩
#align multiplicity.eq_coe_iff multiplicity.eq_coe_iff
theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
exact
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
fun n => h _,
fun h n => h _⟩
#align multiplicity.eq_top_iff multiplicity.eq_top_iff
@[simp]
theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ :=
eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _)
#align multiplicity.is_unit_left multiplicity.isUnit_left
-- @[simp] Porting note: simp can prove this
theorem one_left (b : α) : multiplicity 1 b = ⊤ :=
isUnit_left b isUnit_one
#align multiplicity.one_left multiplicity.one_left
@[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by
rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero]
simp [not_dvd_one_of_finite_one_right ha]
#align multiplicity.get_one_right multiplicity.get_one_right
-- @[simp] Porting note: simp can prove this
theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ :=
isUnit_left a u.isUnit
#align multiplicity.unit_left multiplicity.unit_left
theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by
rw [← Nat.cast_zero, eq_coe_iff]
simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and]
#align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero
theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b :=
multiplicity_eq_zero.not_left
#align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero
theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b :=
Part.eq_none_iff'
#align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite
theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by
rw [Ne.def, eq_top_iff_not_finite, Classical.not_not]
#align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite
theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by
rw [lt_top_iff_ne_top, ne_top_iff_finite]
#align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite
theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩
exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁
#align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd
theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else by
have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _)
rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩
#align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff
theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d :=
⟨fun h n =>
⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩,
fun h =>
le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp)
(multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩
#align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff
theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) :=
multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f
theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b :=
multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f
theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right
theorem eq_of_associated_right {a b c : α} (h : Associated b c) :
multiplicity a b = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd)
(multiplicity_le_multiplicity_of_dvd_right h.symm.dvd)
#align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right
theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
rw [← pow_one a]
apply pow_dvd_of_le_multiplicity
simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h
#align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos
theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos, fun hdvd =>
lt_of_le_of_ne (zero_le _) fun heq =>
is_greatest
(show multiplicity a b < ↑1 by
simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one)
(by rwa [pow_one a])⟩
#align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos
theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1
Classical.by_contradiction fun ha1 : a ≠ 1 =>
have ha_gt_one : 1 < a :=
lt_of_not_ge fun _ =>
match a with
| 0 => ha rfl
| 1 => ha1 rfl
| b+2 => by linarith
not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b),
fun h => by cases h <;> simp [*]⟩
#align multiplicity.finite_nat_iff multiplicity.finite_nat_iff
alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos
end Monoid
section CommMonoid
variable [CommMonoid α]
theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by
rw [mul_comm]; exact finite_of_finite_mul_right
#align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by
rw [pow_one]
exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩
#align multiplicity.is_unit_right multiplicity.isUnit_right
theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 :=
isUnit_right ha isUnit_one
#align multiplicity.one_right multiplicity.one_right
theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 :=
isUnit_right ha u.isUnit
#align multiplicity.unit_right multiplicity.unit_right
open Classical
theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left
theorem eq_of_associated_left {a b c : α} (h : Associated a b) :
multiplicity b c = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd)
(multiplicity_le_multiplicity_of_dvd_left h.symm.dvd)
#align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left
-- Porting note: this was doing nothing in mathlib3 also
-- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos
end CommMonoid
section MonoidWithZero
variable [MonoidWithZero α]
theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h
fun hb => by simp [hb] at hn
#align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem zero (a : α) : multiplicity a 0 = ⊤ :=
Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _)
#align multiplicity.zero multiplicity.zero
@[simp]
theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 :=
multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha
#align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero
end MonoidWithZero
section CommMonoidWithZero
variable [CommMonoidWithZero α]
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
(show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;>
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
· exact pow_multiplicity_dvd h
· exact is_greatest
((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
· suffices ¬Finite (Associates.mk a) (Associates.mk b) by
rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
rw [h, this]
refine'
not_finite_iff_forall.mpr fun n => by
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
exact not_finite_iff_forall.mp h n
#align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity
end CommMonoidWithZero
section Semiring
variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
fun h => by
rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn
#align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add
end Semiring
section Ring
variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ =>
PartENat.natCast_inj.1 (by
rw [PartENat.natCast_get]
exact Eq.symm
(unique (pow_multiplicity_dvd _).neg_right
(mt dvd_neg.1 (is_greatest' _ (lt_succ_self _)))))
#align multiplicity.neg multiplicity.neg
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h]
· rw [Int.coe_nat_multiplicity]
· rw [multiplicity.neg, Int.coe_nat_multiplicity]
#align multiplicity.int.nat_abs multiplicity.Int.natAbs
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
intro h_dvd
· apply multiplicity.is_greatest _ h_dvd
rw [hk, ← Nat.succ_eq_add_one]
norm_cast
apply Nat.lt_succ_self k
· rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
exact PartENat.add_one_le_of_lt h
· have := @min_le_multiplicity_add α _ _ p a b
rwa [← min_eq_right (le_of_lt h)]
#align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption
#align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
· contradiction
· rw [multiplicity_add_of_gt hab, min_eq_right]
exact le_of_lt hab
#align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min
end Ring
section CancelCommMonoidWithZero
variable [CancelCommMonoidWithZero α]
/- Porting note: removed previous wf recursion hints and added termination_by
Also pulled a b intro parameters since Lean parses that more easily -/
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>
have hn0 : 0 < n :=
Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
finite_mul_aux hp hpx hb
⟨s, mul_right_cancel₀ hp.1 (by
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
fun ⟨x, hx⟩ =>
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
hb
(hx.symm ▸
⟨y,
mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
finite_mul_aux hp ha hpx
⟨s, mul_right_cancel₀ hp.1 (by
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩
termination_by finite_mul_aux _ _ n m => n+m
#align multiplicity.finite_mul_aux multiplicity.finite_mul_aux
theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) :=
fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩
#align multiplicity.finite_mul multiplicity.finite_mul
theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b :=
⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h =>
finite_mul hp h.1 h.2⟩
#align multiplicity.finite_mul_iff multiplicity.finite_mul_iff
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
#align multiplicity.finite_pow multiplicity.finite_pow
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩
#align multiplicity.multiplicity_self multiplicity.multiplicity_self
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact
mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩)
#align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
have hpoweq :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
by simp [pow_add]
have hdiv :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
a * b :=
by rw [hpoweq]; apply mul_dvd_mul <;> assumption
have hsucc :
¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 +
1) ∣
a * b :=
fun h =>
not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩
#align multiplicity.mul' multiplicity.mul'
open Classical
protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add,
PartENat.natCast_inj, multiplicity.mul' hp]
else by
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h]
#align multiplicity.mul multiplicity.mul
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
· simp [has, ← ih]
convert multiplicity.mul hp
#align multiplicity.finset.prod multiplicity.Finset.prod
-- Porting note: with protected could not use pow' k in the succ branch
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
rw [succ_eq_add_one, get_eq_get_of_eq _ _ this,
multiplicity.mul' hp, hk, add_mul, one_mul, add_comm]
#align multiplicity.pow' multiplicity.pow'
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
#align multiplicity.pow multiplicity.pow
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
apply Nat.not_succ_le_self
#align multiplicity.multiplicity_pow_self multiplicity.multiplicity_pow_self
theorem multiplicity_pow_self_of_prime {p : α} (hp : Prime p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
multiplicity_pow_self hp.ne_zero hp.not_unit n
#align multiplicity.multiplicity_pow_self_of_prime multiplicity.multiplicity_pow_self_of_prime
end CancelCommMonoidWithZero
end multiplicity
section Nat
open multiplicity
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
intro h
|
have := Nat.dvd_gcd h (hle _ h)
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
intro h
|
Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0
|
Mathlib_RingTheory_Multiplicity
|
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
h : p ^ 1 ∣ a
this : p ^ 1 ∣ gcd a b
⊢ False
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
/-!
# Multiplicity of a divisor
For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves
several basic results on it.
## Main definitions
* `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest
number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers
`n`.
* `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite.
-/
variable {α β : Type*}
open Nat Part
open BigOperators
/-- `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤`-/
def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat :=
PartENat.find fun n => ¬a ^ (n + 1) ∣ b
#align multiplicity multiplicity
namespace multiplicity
section Monoid
variable [Monoid α] [Monoid β]
/-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/
@[reducible]
def Finite (a b : α) : Prop :=
∃ n : ℕ, ¬a ^ (n + 1) ∣ b
#align multiplicity.finite multiplicity.Finite
theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} :
Finite a b ↔ (multiplicity a b).Dom :=
Iff.rfl
#align multiplicity.finite_iff_dom multiplicity.finite_iff_dom
theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b :=
Iff.rfl
#align multiplicity.finite_def multiplicity.finite_def
theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ =>
hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩
#align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right
@[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
simp
#align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity
theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by simpa [Finite, Classical.not_not] using h),
by simp [Finite, multiplicity, Classical.not_not]; tauto⟩
#align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall
theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a :=
let ⟨n, hn⟩ := h
hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1)
#align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite
theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ =>
⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩
#align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right
variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)]
theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
Nat.casesOn k
(fun _ => by
rw [_root_.pow_zero]
exact one_dvd _)
fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk
#align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity
theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get])
#align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd
theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by
rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h)
#align multiplicity.is_greatest multiplicity.is_greatest
theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm)
#align multiplicity.is_greatest' multiplicity.is_greatest'
theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by
refine' zero_lt_iff.2 fun h => _
simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h)
#align multiplicity.pos_of_dvd multiplicity.pos_of_dvd
theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
have : Finite a b := ⟨k, hsucc⟩
rw [PartENat.le_coe_iff]
exact ⟨this, Nat.find_min' _ hsucc⟩
#align multiplicity.unique multiplicity.unique
theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ := by
rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc]
#align multiplicity.unique' multiplicity.unique'
theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) :
(k : PartENat) ≤ multiplicity a b :=
le_of_not_gt fun hk' => is_greatest hk' hk
#align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd
theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} :
a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b :=
⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩
#align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity
theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le]
#align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd
theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
rw [PartENat.lt_coe_iff]
exact ⟨h₁, lt_succ_self _⟩)⟩,
fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩
#align multiplicity.eq_coe_iff multiplicity.eq_coe_iff
theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
exact
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
fun n => h _,
fun h n => h _⟩
#align multiplicity.eq_top_iff multiplicity.eq_top_iff
@[simp]
theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ :=
eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _)
#align multiplicity.is_unit_left multiplicity.isUnit_left
-- @[simp] Porting note: simp can prove this
theorem one_left (b : α) : multiplicity 1 b = ⊤ :=
isUnit_left b isUnit_one
#align multiplicity.one_left multiplicity.one_left
@[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by
rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero]
simp [not_dvd_one_of_finite_one_right ha]
#align multiplicity.get_one_right multiplicity.get_one_right
-- @[simp] Porting note: simp can prove this
theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ :=
isUnit_left a u.isUnit
#align multiplicity.unit_left multiplicity.unit_left
theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by
rw [← Nat.cast_zero, eq_coe_iff]
simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and]
#align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero
theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b :=
multiplicity_eq_zero.not_left
#align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero
theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b :=
Part.eq_none_iff'
#align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite
theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by
rw [Ne.def, eq_top_iff_not_finite, Classical.not_not]
#align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite
theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by
rw [lt_top_iff_ne_top, ne_top_iff_finite]
#align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite
theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩
exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁
#align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd
theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else by
have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _)
rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩
#align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff
theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d :=
⟨fun h n =>
⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩,
fun h =>
le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp)
(multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩
#align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff
theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) :=
multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f
theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b :=
multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f
theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right
theorem eq_of_associated_right {a b c : α} (h : Associated b c) :
multiplicity a b = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd)
(multiplicity_le_multiplicity_of_dvd_right h.symm.dvd)
#align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right
theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
rw [← pow_one a]
apply pow_dvd_of_le_multiplicity
simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h
#align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos
theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos, fun hdvd =>
lt_of_le_of_ne (zero_le _) fun heq =>
is_greatest
(show multiplicity a b < ↑1 by
simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one)
(by rwa [pow_one a])⟩
#align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos
theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1
Classical.by_contradiction fun ha1 : a ≠ 1 =>
have ha_gt_one : 1 < a :=
lt_of_not_ge fun _ =>
match a with
| 0 => ha rfl
| 1 => ha1 rfl
| b+2 => by linarith
not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b),
fun h => by cases h <;> simp [*]⟩
#align multiplicity.finite_nat_iff multiplicity.finite_nat_iff
alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos
end Monoid
section CommMonoid
variable [CommMonoid α]
theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by
rw [mul_comm]; exact finite_of_finite_mul_right
#align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by
rw [pow_one]
exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩
#align multiplicity.is_unit_right multiplicity.isUnit_right
theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 :=
isUnit_right ha isUnit_one
#align multiplicity.one_right multiplicity.one_right
theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 :=
isUnit_right ha u.isUnit
#align multiplicity.unit_right multiplicity.unit_right
open Classical
theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left
theorem eq_of_associated_left {a b c : α} (h : Associated a b) :
multiplicity b c = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd)
(multiplicity_le_multiplicity_of_dvd_left h.symm.dvd)
#align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left
-- Porting note: this was doing nothing in mathlib3 also
-- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos
end CommMonoid
section MonoidWithZero
variable [MonoidWithZero α]
theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h
fun hb => by simp [hb] at hn
#align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem zero (a : α) : multiplicity a 0 = ⊤ :=
Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _)
#align multiplicity.zero multiplicity.zero
@[simp]
theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 :=
multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha
#align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero
end MonoidWithZero
section CommMonoidWithZero
variable [CommMonoidWithZero α]
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
(show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;>
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
· exact pow_multiplicity_dvd h
· exact is_greatest
((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
· suffices ¬Finite (Associates.mk a) (Associates.mk b) by
rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
rw [h, this]
refine'
not_finite_iff_forall.mpr fun n => by
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
exact not_finite_iff_forall.mp h n
#align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity
end CommMonoidWithZero
section Semiring
variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
fun h => by
rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn
#align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add
end Semiring
section Ring
variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ =>
PartENat.natCast_inj.1 (by
rw [PartENat.natCast_get]
exact Eq.symm
(unique (pow_multiplicity_dvd _).neg_right
(mt dvd_neg.1 (is_greatest' _ (lt_succ_self _)))))
#align multiplicity.neg multiplicity.neg
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h]
· rw [Int.coe_nat_multiplicity]
· rw [multiplicity.neg, Int.coe_nat_multiplicity]
#align multiplicity.int.nat_abs multiplicity.Int.natAbs
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
intro h_dvd
· apply multiplicity.is_greatest _ h_dvd
rw [hk, ← Nat.succ_eq_add_one]
norm_cast
apply Nat.lt_succ_self k
· rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
exact PartENat.add_one_le_of_lt h
· have := @min_le_multiplicity_add α _ _ p a b
rwa [← min_eq_right (le_of_lt h)]
#align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption
#align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
· contradiction
· rw [multiplicity_add_of_gt hab, min_eq_right]
exact le_of_lt hab
#align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min
end Ring
section CancelCommMonoidWithZero
variable [CancelCommMonoidWithZero α]
/- Porting note: removed previous wf recursion hints and added termination_by
Also pulled a b intro parameters since Lean parses that more easily -/
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>
have hn0 : 0 < n :=
Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
finite_mul_aux hp hpx hb
⟨s, mul_right_cancel₀ hp.1 (by
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
fun ⟨x, hx⟩ =>
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
hb
(hx.symm ▸
⟨y,
mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
finite_mul_aux hp ha hpx
⟨s, mul_right_cancel₀ hp.1 (by
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩
termination_by finite_mul_aux _ _ n m => n+m
#align multiplicity.finite_mul_aux multiplicity.finite_mul_aux
theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) :=
fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩
#align multiplicity.finite_mul multiplicity.finite_mul
theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b :=
⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h =>
finite_mul hp h.1 h.2⟩
#align multiplicity.finite_mul_iff multiplicity.finite_mul_iff
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
#align multiplicity.finite_pow multiplicity.finite_pow
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩
#align multiplicity.multiplicity_self multiplicity.multiplicity_self
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact
mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩)
#align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
have hpoweq :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
by simp [pow_add]
have hdiv :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
a * b :=
by rw [hpoweq]; apply mul_dvd_mul <;> assumption
have hsucc :
¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 +
1) ∣
a * b :=
fun h =>
not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩
#align multiplicity.mul' multiplicity.mul'
open Classical
protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add,
PartENat.natCast_inj, multiplicity.mul' hp]
else by
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h]
#align multiplicity.mul multiplicity.mul
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
· simp [has, ← ih]
convert multiplicity.mul hp
#align multiplicity.finset.prod multiplicity.Finset.prod
-- Porting note: with protected could not use pow' k in the succ branch
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
rw [succ_eq_add_one, get_eq_get_of_eq _ _ this,
multiplicity.mul' hp, hk, add_mul, one_mul, add_comm]
#align multiplicity.pow' multiplicity.pow'
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
#align multiplicity.pow multiplicity.pow
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
apply Nat.not_succ_le_self
#align multiplicity.multiplicity_pow_self multiplicity.multiplicity_pow_self
theorem multiplicity_pow_self_of_prime {p : α} (hp : Prime p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
multiplicity_pow_self hp.ne_zero hp.not_unit n
#align multiplicity.multiplicity_pow_self_of_prime multiplicity.multiplicity_pow_self_of_prime
end CancelCommMonoidWithZero
end multiplicity
section Nat
open multiplicity
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
intro h
have := Nat.dvd_gcd h (hle _ h)
|
rw [Coprime.gcd_eq_one hab, Nat.dvd_one, pow_one] at this
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
intro h
have := Nat.dvd_gcd h (hle _ h)
|
Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0
|
Mathlib_RingTheory_Multiplicity
|
α : Type u_1
β : Type u_2
p a b : ℕ
hp : p ≠ 1
hle : ∀ (n : ℕ), p ^ n ∣ a → p ^ n ∣ b
hab : Coprime a b
h : p ^ 1 ∣ a
this : p = 1
⊢ False
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
/-!
# Multiplicity of a divisor
For a commutative monoid, this file introduces the notion of multiplicity of a divisor and proves
several basic results on it.
## Main definitions
* `multiplicity a b`: for two elements `a` and `b` of a commutative monoid returns the largest
number `n` such that `a ^ n ∣ b` or infinity, written `⊤`, if `a ^ n ∣ b` for all natural numbers
`n`.
* `multiplicity.Finite a b`: a predicate denoting that the multiplicity of `a` in `b` is finite.
-/
variable {α β : Type*}
open Nat Part
open BigOperators
/-- `multiplicity a b` returns the largest natural number `n` such that
`a ^ n ∣ b`, as a `PartENat` or natural with infinity. If `∀ n, a ^ n ∣ b`,
then it returns `⊤`-/
def multiplicity [Monoid α] [DecidableRel ((· ∣ ·) : α → α → Prop)] (a b : α) : PartENat :=
PartENat.find fun n => ¬a ^ (n + 1) ∣ b
#align multiplicity multiplicity
namespace multiplicity
section Monoid
variable [Monoid α] [Monoid β]
/-- `multiplicity.Finite a b` indicates that the multiplicity of `a` in `b` is finite. -/
@[reducible]
def Finite (a b : α) : Prop :=
∃ n : ℕ, ¬a ^ (n + 1) ∣ b
#align multiplicity.finite multiplicity.Finite
theorem finite_iff_dom [DecidableRel ((· ∣ ·) : α → α → Prop)] {a b : α} :
Finite a b ↔ (multiplicity a b).Dom :=
Iff.rfl
#align multiplicity.finite_iff_dom multiplicity.finite_iff_dom
theorem finite_def {a b : α} : Finite a b ↔ ∃ n : ℕ, ¬a ^ (n + 1) ∣ b :=
Iff.rfl
#align multiplicity.finite_def multiplicity.finite_def
theorem not_dvd_one_of_finite_one_right {a : α} : Finite a 1 → ¬a ∣ 1 := fun ⟨n, hn⟩ ⟨d, hd⟩ =>
hn ⟨d ^ (n + 1), (pow_mul_pow_eq_one (n + 1) hd.symm).symm⟩
#align multiplicity.not_dvd_one_of_finite_one_right multiplicity.not_dvd_one_of_finite_one_right
@[norm_cast]
theorem Int.coe_nat_multiplicity (a b : ℕ) : multiplicity (a : ℤ) (b : ℤ) = multiplicity a b := by
apply Part.ext'
· rw [← @finite_iff_dom ℕ, @finite_def ℕ, ← @finite_iff_dom ℤ, @finite_def ℤ]
norm_cast
· intro h1 h2
apply _root_.le_antisymm <;>
· apply Nat.find_mono
norm_cast
simp
#align multiplicity.int.coe_nat_multiplicity multiplicity.Int.coe_nat_multiplicity
theorem not_finite_iff_forall {a b : α} : ¬Finite a b ↔ ∀ n : ℕ, a ^ n ∣ b :=
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
(by simpa [Finite, Classical.not_not] using h),
by simp [Finite, multiplicity, Classical.not_not]; tauto⟩
#align multiplicity.not_finite_iff_forall multiplicity.not_finite_iff_forall
theorem not_unit_of_finite {a b : α} (h : Finite a b) : ¬IsUnit a :=
let ⟨n, hn⟩ := h
hn ∘ IsUnit.dvd ∘ IsUnit.pow (n + 1)
#align multiplicity.not_unit_of_finite multiplicity.not_unit_of_finite
theorem finite_of_finite_mul_right {a b c : α} : Finite a (b * c) → Finite a b := fun ⟨n, hn⟩ =>
⟨n, fun h => hn (h.trans (dvd_mul_right _ _))⟩
#align multiplicity.finite_of_finite_mul_right multiplicity.finite_of_finite_mul_right
variable [DecidableRel ((· ∣ ·) : α → α → Prop)] [DecidableRel ((· ∣ ·) : β → β → Prop)]
theorem pow_dvd_of_le_multiplicity {a b : α} {k : ℕ} :
(k : PartENat) ≤ multiplicity a b → a ^ k ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
Nat.casesOn k
(fun _ => by
rw [_root_.pow_zero]
exact one_dvd _)
fun k ⟨_, h₂⟩ => by_contradiction fun hk => Nat.find_min _ (lt_of_succ_le (h₂ ⟨k, hk⟩)) hk
#align multiplicity.pow_dvd_of_le_multiplicity multiplicity.pow_dvd_of_le_multiplicity
theorem pow_multiplicity_dvd {a b : α} (h : Finite a b) : a ^ get (multiplicity a b) h ∣ b :=
pow_dvd_of_le_multiplicity (by rw [PartENat.natCast_get])
#align multiplicity.pow_multiplicity_dvd multiplicity.pow_multiplicity_dvd
theorem is_greatest {a b : α} {m : ℕ} (hm : multiplicity a b < m) : ¬a ^ m ∣ b := fun h => by
rw [PartENat.lt_coe_iff] at hm; exact Nat.find_spec hm.fst ((pow_dvd_pow _ hm.snd).trans h)
#align multiplicity.is_greatest multiplicity.is_greatest
theorem is_greatest' {a b : α} {m : ℕ} (h : Finite a b) (hm : get (multiplicity a b) h < m) :
¬a ^ m ∣ b :=
is_greatest (by rwa [← PartENat.coe_lt_coe, PartENat.natCast_get] at hm)
#align multiplicity.is_greatest' multiplicity.is_greatest'
theorem pos_of_dvd {a b : α} (hfin : Finite a b) (hdiv : a ∣ b) :
0 < (multiplicity a b).get hfin := by
refine' zero_lt_iff.2 fun h => _
simpa [hdiv] using is_greatest' hfin (lt_one_iff.mpr h)
#align multiplicity.pos_of_dvd multiplicity.pos_of_dvd
theorem unique {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
(k : PartENat) = multiplicity a b :=
le_antisymm (le_of_not_gt fun hk' => is_greatest hk' hk) <| by
have : Finite a b := ⟨k, hsucc⟩
rw [PartENat.le_coe_iff]
exact ⟨this, Nat.find_min' _ hsucc⟩
#align multiplicity.unique multiplicity.unique
theorem unique' {a b : α} {k : ℕ} (hk : a ^ k ∣ b) (hsucc : ¬a ^ (k + 1) ∣ b) :
k = get (multiplicity a b) ⟨k, hsucc⟩ := by
rw [← PartENat.natCast_inj, PartENat.natCast_get, unique hk hsucc]
#align multiplicity.unique' multiplicity.unique'
theorem le_multiplicity_of_pow_dvd {a b : α} {k : ℕ} (hk : a ^ k ∣ b) :
(k : PartENat) ≤ multiplicity a b :=
le_of_not_gt fun hk' => is_greatest hk' hk
#align multiplicity.le_multiplicity_of_pow_dvd multiplicity.le_multiplicity_of_pow_dvd
theorem pow_dvd_iff_le_multiplicity {a b : α} {k : ℕ} :
a ^ k ∣ b ↔ (k : PartENat) ≤ multiplicity a b :=
⟨le_multiplicity_of_pow_dvd, pow_dvd_of_le_multiplicity⟩
#align multiplicity.pow_dvd_iff_le_multiplicity multiplicity.pow_dvd_iff_le_multiplicity
theorem multiplicity_lt_iff_not_dvd {a b : α} {k : ℕ} :
multiplicity a b < (k : PartENat) ↔ ¬a ^ k ∣ b := by rw [pow_dvd_iff_le_multiplicity, not_le]
#align multiplicity.multiplicity_lt_iff_neg_dvd multiplicity.multiplicity_lt_iff_not_dvd
theorem eq_coe_iff {a b : α} {n : ℕ} :
multiplicity a b = (n : PartENat) ↔ a ^ n ∣ b ∧ ¬a ^ (n + 1) ∣ b := by
rw [← PartENat.some_eq_natCast]
exact
⟨fun h =>
let ⟨h₁, h₂⟩ := eq_some_iff.1 h
h₂ ▸ ⟨pow_multiplicity_dvd _, is_greatest (by
rw [PartENat.lt_coe_iff]
exact ⟨h₁, lt_succ_self _⟩)⟩,
fun h => eq_some_iff.2 ⟨⟨n, h.2⟩, Eq.symm <| unique' h.1 h.2⟩⟩
#align multiplicity.eq_coe_iff multiplicity.eq_coe_iff
theorem eq_top_iff {a b : α} : multiplicity a b = ⊤ ↔ ∀ n : ℕ, a ^ n ∣ b :=
(PartENat.find_eq_top_iff _).trans <| by
simp only [Classical.not_not]
exact
⟨fun h n =>
Nat.casesOn n
(by
rw [_root_.pow_zero]
exact one_dvd _)
fun n => h _,
fun h n => h _⟩
#align multiplicity.eq_top_iff multiplicity.eq_top_iff
@[simp]
theorem isUnit_left {a : α} (b : α) (ha : IsUnit a) : multiplicity a b = ⊤ :=
eq_top_iff.2 fun _ => IsUnit.dvd (ha.pow _)
#align multiplicity.is_unit_left multiplicity.isUnit_left
-- @[simp] Porting note: simp can prove this
theorem one_left (b : α) : multiplicity 1 b = ⊤ :=
isUnit_left b isUnit_one
#align multiplicity.one_left multiplicity.one_left
@[simp]
theorem get_one_right {a : α} (ha : Finite a 1) : get (multiplicity a 1) ha = 0 := by
rw [PartENat.get_eq_iff_eq_coe, eq_coe_iff, _root_.pow_zero]
simp [not_dvd_one_of_finite_one_right ha]
#align multiplicity.get_one_right multiplicity.get_one_right
-- @[simp] Porting note: simp can prove this
theorem unit_left (a : α) (u : αˣ) : multiplicity (u : α) a = ⊤ :=
isUnit_left a u.isUnit
#align multiplicity.unit_left multiplicity.unit_left
theorem multiplicity_eq_zero {a b : α} : multiplicity a b = 0 ↔ ¬a ∣ b := by
rw [← Nat.cast_zero, eq_coe_iff]
simp only [_root_.pow_zero, isUnit_one, IsUnit.dvd, zero_add, pow_one, true_and]
#align multiplicity.multiplicity_eq_zero multiplicity.multiplicity_eq_zero
theorem multiplicity_ne_zero {a b : α} : multiplicity a b ≠ 0 ↔ a ∣ b :=
multiplicity_eq_zero.not_left
#align multiplicity.multiplicity_ne_zero multiplicity.multiplicity_ne_zero
theorem eq_top_iff_not_finite {a b : α} : multiplicity a b = ⊤ ↔ ¬Finite a b :=
Part.eq_none_iff'
#align multiplicity.eq_top_iff_not_finite multiplicity.eq_top_iff_not_finite
theorem ne_top_iff_finite {a b : α} : multiplicity a b ≠ ⊤ ↔ Finite a b := by
rw [Ne.def, eq_top_iff_not_finite, Classical.not_not]
#align multiplicity.ne_top_iff_finite multiplicity.ne_top_iff_finite
theorem lt_top_iff_finite {a b : α} : multiplicity a b < ⊤ ↔ Finite a b := by
rw [lt_top_iff_ne_top, ne_top_iff_finite]
#align multiplicity.lt_top_iff_finite multiplicity.lt_top_iff_finite
theorem exists_eq_pow_mul_and_not_dvd {a b : α} (hfin : Finite a b) :
∃ c : α, b = a ^ (multiplicity a b).get hfin * c ∧ ¬a ∣ c := by
obtain ⟨c, hc⟩ := multiplicity.pow_multiplicity_dvd hfin
refine' ⟨c, hc, _⟩
rintro ⟨k, hk⟩
rw [hk, ← mul_assoc, ← _root_.pow_succ'] at hc
have h₁ : a ^ ((multiplicity a b).get hfin + 1) ∣ b := ⟨k, hc⟩
exact (multiplicity.eq_coe_iff.1 (by simp)).2 h₁
#align multiplicity.exists_eq_pow_mul_and_not_dvd multiplicity.exists_eq_pow_mul_and_not_dvd
theorem multiplicity_le_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b ≤ multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b → c ^ n ∣ d :=
⟨fun h n hab => pow_dvd_of_le_multiplicity (le_trans (le_multiplicity_of_pow_dvd hab) h), fun h =>
letI := Classical.dec (Finite a b)
if hab : Finite a b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 hab)];
exact le_multiplicity_of_pow_dvd (h _ (pow_multiplicity_dvd _))
else by
have : ∀ n : ℕ, c ^ n ∣ d := fun n => h n (not_finite_iff_forall.1 hab _)
rw [eq_top_iff_not_finite.2 hab, eq_top_iff_not_finite.2 (not_finite_iff_forall.2 this)]⟩
#align multiplicity.multiplicity_le_multiplicity_iff multiplicity.multiplicity_le_multiplicity_iff
theorem multiplicity_eq_multiplicity_iff {a b : α} {c d : β} :
multiplicity a b = multiplicity c d ↔ ∀ n : ℕ, a ^ n ∣ b ↔ c ^ n ∣ d :=
⟨fun h n =>
⟨multiplicity_le_multiplicity_iff.mp h.le n, multiplicity_le_multiplicity_iff.mp h.ge n⟩,
fun h =>
le_antisymm (multiplicity_le_multiplicity_iff.mpr fun n => (h n).mp)
(multiplicity_le_multiplicity_iff.mpr fun n => (h n).mpr)⟩
#align multiplicity.multiplicity_eq_multiplicity_iff multiplicity.multiplicity_eq_multiplicity_iff
theorem le_multiplicity_map {F : Type*} [MonoidHomClass F α β] (f : F) {a b : α} :
multiplicity a b ≤ multiplicity (f a) (f b) :=
multiplicity_le_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd f
theorem multiplicity_map_eq {F : Type*} [MulEquivClass F α β] (f : F) {a b : α} :
multiplicity (f a) (f b) = multiplicity a b :=
multiplicity_eq_multiplicity_iff.mpr fun n ↦ by rw [← map_pow]; exact map_dvd_iff f
theorem multiplicity_le_multiplicity_of_dvd_right {a b c : α} (h : b ∣ c) :
multiplicity a b ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun _ hb => hb.trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_right multiplicity.multiplicity_le_multiplicity_of_dvd_right
theorem eq_of_associated_right {a b c : α} (h : Associated b c) :
multiplicity a b = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_right h.dvd)
(multiplicity_le_multiplicity_of_dvd_right h.symm.dvd)
#align multiplicity.eq_of_associated_right multiplicity.eq_of_associated_right
theorem dvd_of_multiplicity_pos {a b : α} (h : (0 : PartENat) < multiplicity a b) : a ∣ b := by
rw [← pow_one a]
apply pow_dvd_of_le_multiplicity
simpa only [Nat.cast_one, PartENat.pos_iff_one_le] using h
#align multiplicity.dvd_of_multiplicity_pos multiplicity.dvd_of_multiplicity_pos
theorem dvd_iff_multiplicity_pos {a b : α} : (0 : PartENat) < multiplicity a b ↔ a ∣ b :=
⟨dvd_of_multiplicity_pos, fun hdvd =>
lt_of_le_of_ne (zero_le _) fun heq =>
is_greatest
(show multiplicity a b < ↑1 by
simpa only [heq, Nat.cast_zero] using PartENat.coe_lt_coe.mpr zero_lt_one)
(by rwa [pow_one a])⟩
#align multiplicity.dvd_iff_multiplicity_pos multiplicity.dvd_iff_multiplicity_pos
theorem finite_nat_iff {a b : ℕ} : Finite a b ↔ a ≠ 1 ∧ 0 < b := by
rw [← not_iff_not, not_finite_iff_forall, not_and_or, Ne.def, Classical.not_not, not_lt,
le_zero_iff]
exact
⟨fun h =>
or_iff_not_imp_right.2 fun hb =>
have ha : a ≠ 0 := fun ha => hb <| zero_dvd_iff.mp <| by rw [ha] at h; exact h 1
Classical.by_contradiction fun ha1 : a ≠ 1 =>
have ha_gt_one : 1 < a :=
lt_of_not_ge fun _ =>
match a with
| 0 => ha rfl
| 1 => ha1 rfl
| b+2 => by linarith
not_lt_of_ge (le_of_dvd (Nat.pos_of_ne_zero hb) (h b)) (lt_pow_self ha_gt_one b),
fun h => by cases h <;> simp [*]⟩
#align multiplicity.finite_nat_iff multiplicity.finite_nat_iff
alias ⟨_, _root_.has_dvd.dvd.multiplicity_pos⟩ := dvd_iff_multiplicity_pos
end Monoid
section CommMonoid
variable [CommMonoid α]
theorem finite_of_finite_mul_left {a b c : α} : Finite a (b * c) → Finite a c := by
rw [mul_comm]; exact finite_of_finite_mul_right
#align multiplicity.finite_of_finite_mul_left multiplicity.finite_of_finite_mul_left
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem isUnit_right {a b : α} (ha : ¬IsUnit a) (hb : IsUnit b) : multiplicity a b = 0 :=
eq_coe_iff.2
⟨show a ^ 0 ∣ b by simp only [_root_.pow_zero, one_dvd], by
rw [pow_one]
exact fun h => mt (isUnit_of_dvd_unit h) ha hb⟩
#align multiplicity.is_unit_right multiplicity.isUnit_right
theorem one_right {a : α} (ha : ¬IsUnit a) : multiplicity a 1 = 0 :=
isUnit_right ha isUnit_one
#align multiplicity.one_right multiplicity.one_right
theorem unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 :=
isUnit_right ha u.isUnit
#align multiplicity.unit_right multiplicity.unit_right
open Classical
theorem multiplicity_le_multiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
multiplicity b c ≤ multiplicity a c :=
multiplicity_le_multiplicity_iff.2 fun n h => (pow_dvd_pow_of_dvd hdvd n).trans h
#align multiplicity.multiplicity_le_multiplicity_of_dvd_left multiplicity.multiplicity_le_multiplicity_of_dvd_left
theorem eq_of_associated_left {a b c : α} (h : Associated a b) :
multiplicity b c = multiplicity a c :=
le_antisymm (multiplicity_le_multiplicity_of_dvd_left h.dvd)
(multiplicity_le_multiplicity_of_dvd_left h.symm.dvd)
#align multiplicity.eq_of_associated_left multiplicity.eq_of_associated_left
-- Porting note: this was doing nothing in mathlib3 also
-- alias dvd_iff_multiplicity_pos ↔ _ _root_.has_dvd.dvd.multiplicity_pos
end CommMonoid
section MonoidWithZero
variable [MonoidWithZero α]
theorem ne_zero_of_finite {a b : α} (h : Finite a b) : b ≠ 0 :=
let ⟨n, hn⟩ := h
fun hb => by simp [hb] at hn
#align multiplicity.ne_zero_of_finite multiplicity.ne_zero_of_finite
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem zero (a : α) : multiplicity a 0 = ⊤ :=
Part.eq_none_iff.2 fun _ ⟨⟨_, hk⟩, _⟩ => hk (dvd_zero _)
#align multiplicity.zero multiplicity.zero
@[simp]
theorem multiplicity_zero_eq_zero_of_ne_zero (a : α) (ha : a ≠ 0) : multiplicity 0 a = 0 :=
multiplicity.multiplicity_eq_zero.2 <| mt zero_dvd_iff.1 ha
#align multiplicity.multiplicity_zero_eq_zero_of_ne_zero multiplicity.multiplicity_zero_eq_zero_of_ne_zero
end MonoidWithZero
section CommMonoidWithZero
variable [CommMonoidWithZero α]
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem multiplicity_mk_eq_multiplicity
[DecidableRel ((· ∣ ·) : Associates α → Associates α → Prop)] {a b : α} :
multiplicity (Associates.mk a) (Associates.mk b) = multiplicity a b := by
by_cases h : Finite a b
· rw [← PartENat.natCast_get (finite_iff_dom.mp h)]
refine'
(multiplicity.unique
(show Associates.mk a ^ (multiplicity a b).get h ∣ Associates.mk b from _) _).symm <;>
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
· exact pow_multiplicity_dvd h
· exact is_greatest
((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _)))
· suffices ¬Finite (Associates.mk a) (Associates.mk b) by
rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this
rw [h, this]
refine'
not_finite_iff_forall.mpr fun n => by
rw [← Associates.mk_pow, Associates.mk_dvd_mk]
exact not_finite_iff_forall.mp h n
#align multiplicity.multiplicity_mk_eq_multiplicity multiplicity.multiplicity_mk_eq_multiplicity
end CommMonoidWithZero
section Semiring
variable [Semiring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
theorem min_le_multiplicity_add {p a b : α} :
min (multiplicity p a) (multiplicity p b) ≤ multiplicity p (a + b) :=
(le_total (multiplicity p a) (multiplicity p b)).elim
(fun h => by
rw [min_eq_left h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add hn (multiplicity_le_multiplicity_iff.1 h n hn))
fun h => by
rw [min_eq_right h, multiplicity_le_multiplicity_iff];
exact fun n hn => dvd_add (multiplicity_le_multiplicity_iff.1 h n hn) hn
#align multiplicity.min_le_multiplicity_add multiplicity.min_le_multiplicity_add
end Semiring
section Ring
variable [Ring α] [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
protected theorem neg (a b : α) : multiplicity a (-b) = multiplicity a b :=
Part.ext' (by simp only [multiplicity, PartENat.find, dvd_neg]) fun h₁ h₂ =>
PartENat.natCast_inj.1 (by
rw [PartENat.natCast_get]
exact Eq.symm
(unique (pow_multiplicity_dvd _).neg_right
(mt dvd_neg.1 (is_greatest' _ (lt_succ_self _)))))
#align multiplicity.neg multiplicity.neg
theorem Int.natAbs (a : ℕ) (b : ℤ) : multiplicity a b.natAbs = multiplicity (a : ℤ) b := by
cases' Int.natAbs_eq b with h h <;> conv_rhs => rw [h]
· rw [Int.coe_nat_multiplicity]
· rw [multiplicity.neg, Int.coe_nat_multiplicity]
#align multiplicity.int.nat_abs multiplicity.Int.natAbs
theorem multiplicity_add_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a + b) = multiplicity p b := by
apply le_antisymm
· apply PartENat.le_of_lt_add_one
cases' PartENat.ne_top_iff.mp (PartENat.ne_top_of_lt h) with k hk
rw [hk]
rw_mod_cast [multiplicity_lt_iff_not_dvd, dvd_add_right]
intro h_dvd
· apply multiplicity.is_greatest _ h_dvd
rw [hk, ← Nat.succ_eq_add_one]
norm_cast
apply Nat.lt_succ_self k
· rw [pow_dvd_iff_le_multiplicity, Nat.cast_add, ← hk, Nat.cast_one]
exact PartENat.add_one_le_of_lt h
· have := @min_le_multiplicity_add α _ _ p a b
rwa [← min_eq_right (le_of_lt h)]
#align multiplicity.multiplicity_add_of_gt multiplicity.multiplicity_add_of_gt
theorem multiplicity_sub_of_gt {p a b : α} (h : multiplicity p b < multiplicity p a) :
multiplicity p (a - b) = multiplicity p b := by
rw [sub_eq_add_neg, multiplicity_add_of_gt] <;> rw [multiplicity.neg]; assumption
#align multiplicity.multiplicity_sub_of_gt multiplicity.multiplicity_sub_of_gt
theorem multiplicity_add_eq_min {p a b : α} (h : multiplicity p a ≠ multiplicity p b) :
multiplicity p (a + b) = min (multiplicity p a) (multiplicity p b) := by
rcases lt_trichotomy (multiplicity p a) (multiplicity p b) with (hab | hab | hab)
· rw [add_comm, multiplicity_add_of_gt hab, min_eq_left]
exact le_of_lt hab
· contradiction
· rw [multiplicity_add_of_gt hab, min_eq_right]
exact le_of_lt hab
#align multiplicity.multiplicity_add_eq_min multiplicity.multiplicity_add_eq_min
end Ring
section CancelCommMonoidWithZero
variable [CancelCommMonoidWithZero α]
/- Porting note: removed previous wf recursion hints and added termination_by
Also pulled a b intro parameters since Lean parses that more easily -/
theorem finite_mul_aux {p : α} (hp : Prime p) {a b : α} :
∀ {n m : ℕ}, ¬p ^ (n + 1) ∣ a → ¬p ^ (m + 1) ∣ b → ¬p ^ (n + m + 1) ∣ a * b
| n, m => fun ha hb ⟨s, hs⟩ =>
have : p ∣ a * b := ⟨p ^ (n + m) * s, by simp [hs, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩
(hp.2.2 a b this).elim
(fun ⟨x, hx⟩ =>
have hn0 : 0 < n :=
Nat.pos_of_ne_zero fun hn0 => by simp [hx, hn0] at ha
have hpx : ¬p ^ (n - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
ha (hx.symm ▸ ⟨y, mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hn0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
have : 1 ≤ n + m := le_trans hn0 (Nat.le_add_right n m)
finite_mul_aux hp hpx hb
⟨s, mul_right_cancel₀ hp.1 (by
rw [tsub_add_eq_add_tsub (succ_le_of_lt hn0), tsub_add_cancel_of_le this]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩)
fun ⟨x, hx⟩ =>
have hm0 : 0 < m :=
Nat.pos_of_ne_zero fun hm0 => by simp [hx, hm0] at hb
have hpx : ¬p ^ (m - 1 + 1) ∣ x := fun ⟨y, hy⟩ =>
hb
(hx.symm ▸
⟨y,
mul_right_cancel₀ hp.1 <| by
rw [tsub_add_cancel_of_le (succ_le_of_lt hm0)] at hy;
simp [hy, pow_add, mul_comm, mul_assoc, mul_left_comm]⟩)
finite_mul_aux hp ha hpx
⟨s, mul_right_cancel₀ hp.1 (by
rw [add_assoc, tsub_add_cancel_of_le (succ_le_of_lt hm0)]
simp_all [mul_comm, mul_assoc, mul_left_comm, pow_add])⟩
termination_by finite_mul_aux _ _ n m => n+m
#align multiplicity.finite_mul_aux multiplicity.finite_mul_aux
theorem finite_mul {p a b : α} (hp : Prime p) : Finite p a → Finite p b → Finite p (a * b) :=
fun ⟨n, hn⟩ ⟨m, hm⟩ => ⟨n + m, finite_mul_aux hp hn hm⟩
#align multiplicity.finite_mul multiplicity.finite_mul
theorem finite_mul_iff {p a b : α} (hp : Prime p) : Finite p (a * b) ↔ Finite p a ∧ Finite p b :=
⟨fun h => ⟨finite_of_finite_mul_right h, finite_of_finite_mul_left h⟩, fun h =>
finite_mul hp h.1 h.2⟩
#align multiplicity.finite_mul_iff multiplicity.finite_mul_iff
theorem finite_pow {p a : α} (hp : Prime p) : ∀ {k : ℕ} (_ : Finite p a), Finite p (a ^ k)
| 0, _ => ⟨0, by simp [mt isUnit_iff_dvd_one.2 hp.2.1]⟩
| k + 1, ha => by rw [_root_.pow_succ]; exact finite_mul hp ha (finite_pow hp ha)
#align multiplicity.finite_pow multiplicity.finite_pow
variable [DecidableRel ((· ∣ ·) : α → α → Prop)]
@[simp]
theorem multiplicity_self {a : α} (ha : ¬IsUnit a) (ha0 : a ≠ 0) : multiplicity a a = 1 := by
rw [← Nat.cast_one]
exact eq_coe_iff.2 ⟨by simp, fun ⟨b, hb⟩ => ha (isUnit_iff_dvd_one.2
⟨b, mul_left_cancel₀ ha0 <| by simpa [_root_.pow_succ, mul_assoc] using hb⟩)⟩
#align multiplicity.multiplicity_self multiplicity.multiplicity_self
@[simp]
theorem get_multiplicity_self {a : α} (ha : Finite a a) : get (multiplicity a a) ha = 1 :=
PartENat.get_eq_iff_eq_coe.2
(eq_coe_iff.2
⟨by simp, fun ⟨b, hb⟩ => by
rw [← mul_one a, pow_add, pow_one, mul_assoc, mul_assoc,
mul_right_inj' (ne_zero_of_finite ha)] at hb;
exact
mt isUnit_iff_dvd_one.2 (not_unit_of_finite ha) ⟨b, by simp_all⟩⟩)
#align multiplicity.get_multiplicity_self multiplicity.get_multiplicity_self
protected theorem mul' {p a b : α} (hp : Prime p) (h : (multiplicity p (a * b)).Dom) :
get (multiplicity p (a * b)) h =
get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 := by
have hdiva : p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 ∣ a := pow_multiplicity_dvd _
have hdivb : p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 ∣ b := pow_multiplicity_dvd _
have hpoweq :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) =
p ^ get (multiplicity p a) ((finite_mul_iff hp).1 h).1 *
p ^ get (multiplicity p b) ((finite_mul_iff hp).1 h).2 :=
by simp [pow_add]
have hdiv :
p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2) ∣
a * b :=
by rw [hpoweq]; apply mul_dvd_mul <;> assumption
have hsucc :
¬p ^ (get (multiplicity p a) ((finite_mul_iff hp).1 h).1 +
get (multiplicity p b) ((finite_mul_iff hp).1 h).2 +
1) ∣
a * b :=
fun h =>
not_or_of_not (is_greatest' _ (lt_succ_self _)) (is_greatest' _ (lt_succ_self _))
(_root_.succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul hp hdiva hdivb h)
rw [← PartENat.natCast_inj, PartENat.natCast_get, eq_coe_iff]; exact ⟨hdiv, hsucc⟩
#align multiplicity.mul' multiplicity.mul'
open Classical
protected theorem mul {p a b : α} (hp : Prime p) :
multiplicity p (a * b) = multiplicity p a + multiplicity p b :=
if h : Finite p a ∧ Finite p b then by
rw [← PartENat.natCast_get (finite_iff_dom.1 h.1), ←
PartENat.natCast_get (finite_iff_dom.1 h.2), ←
PartENat.natCast_get (finite_iff_dom.1 (finite_mul hp h.1 h.2)), ← Nat.cast_add,
PartENat.natCast_inj, multiplicity.mul' hp]
else by
rw [eq_top_iff_not_finite.2 (mt (finite_mul_iff hp).1 h)]
cases' not_and_or.1 h with h h <;> simp [eq_top_iff_not_finite.2 h]
#align multiplicity.mul multiplicity.mul
theorem Finset.prod {β : Type*} {p : α} (hp : Prime p) (s : Finset β) (f : β → α) :
multiplicity p (∏ x in s, f x) = ∑ x in s, multiplicity p (f x) := by
classical
induction' s using Finset.induction with a s has ih h
· simp only [Finset.sum_empty, Finset.prod_empty]
convert one_right hp.not_unit
· simp [has, ← ih]
convert multiplicity.mul hp
#align multiplicity.finset.prod multiplicity.Finset.prod
-- Porting note: with protected could not use pow' k in the succ branch
protected theorem pow' {p a : α} (hp : Prime p) (ha : Finite p a) :
∀ {k : ℕ}, get (multiplicity p (a ^ k)) (finite_pow hp ha) = k * get (multiplicity p a) ha := by
intro k
induction' k with k hk
· simp [one_right hp.not_unit]
· have : multiplicity p (a ^ (k + 1)) = multiplicity p (a * a ^ k) := by rw [_root_.pow_succ]
rw [succ_eq_add_one, get_eq_get_of_eq _ _ this,
multiplicity.mul' hp, hk, add_mul, one_mul, add_comm]
#align multiplicity.pow' multiplicity.pow'
theorem pow {p a : α} (hp : Prime p) : ∀ {k : ℕ}, multiplicity p (a ^ k) = k • multiplicity p a
| 0 => by simp [one_right hp.not_unit]
| succ k => by simp [_root_.pow_succ, succ_nsmul, pow hp, multiplicity.mul hp]
#align multiplicity.pow multiplicity.pow
theorem multiplicity_pow_self {p : α} (h0 : p ≠ 0) (hu : ¬IsUnit p) (n : ℕ) :
multiplicity p (p ^ n) = n := by
rw [eq_coe_iff]
use dvd_rfl
rw [pow_dvd_pow_iff h0 hu]
apply Nat.not_succ_le_self
#align multiplicity.multiplicity_pow_self multiplicity.multiplicity_pow_self
theorem multiplicity_pow_self_of_prime {p : α} (hp : Prime p) (n : ℕ) :
multiplicity p (p ^ n) = n :=
multiplicity_pow_self hp.ne_zero hp.not_unit n
#align multiplicity.multiplicity_pow_self_of_prime multiplicity.multiplicity_pow_self_of_prime
end CancelCommMonoidWithZero
end multiplicity
section Nat
open multiplicity
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
intro h
have := Nat.dvd_gcd h (hle _ h)
rw [Coprime.gcd_eq_one hab, Nat.dvd_one, pow_one] at this
|
exact hp this
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0 := by
rw [multiplicity_le_multiplicity_iff] at hle
rw [← nonpos_iff_eq_zero, ← not_lt, PartENat.pos_iff_one_le, ← Nat.cast_one, ←
pow_dvd_iff_le_multiplicity]
intro h
have := Nat.dvd_gcd h (hle _ h)
rw [Coprime.gcd_eq_one hab, Nat.dvd_one, pow_one] at this
|
Mathlib.RingTheory.Multiplicity.640_0.uTHZeAJqYiw3Jx8
|
theorem multiplicity_eq_zero_of_coprime {p a b : ℕ} (hp : p ≠ 1)
(hle : multiplicity p a ≤ multiplicity p b) (hab : Nat.Coprime a b) : multiplicity p a = 0
|
Mathlib_RingTheory_Multiplicity
|
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
h : ¬1 ≤ a
⊢ ¬1 ≤ a * a
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Ordered monoids
This file provides the definitions of ordered monoids.
-/
open Function
variable {α β : Type*}
/-- An ordered (additive) commutative monoid is a commutative monoid with a partial order such that
addition is monotone. -/
class OrderedAddCommMonoid (α : Type*) extends AddCommMonoid α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c, c + a ≤ c + b
#align ordered_add_comm_monoid OrderedAddCommMonoid
/-- An ordered commutative monoid is a commutative monoid with a partial order such that
multiplication is monotone. -/
@[to_additive]
class OrderedCommMonoid (α : Type*) extends CommMonoid α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c, c * a ≤ c * b
#align ordered_comm_monoid OrderedCommMonoid
section OrderedCommMonoid
variable [OrderedCommMonoid α]
@[to_additive]
instance OrderedCommMonoid.toCovariantClassLeft : CovariantClass α α (· * ·) (· ≤ ·) where
elim := fun a _ _ bc ↦ OrderedCommMonoid.mul_le_mul_left _ _ bc a
#align ordered_comm_monoid.to_covariant_class_left OrderedCommMonoid.toCovariantClassLeft
#align ordered_add_comm_monoid.to_covariant_class_left OrderedAddCommMonoid.toCovariantClassLeft
/- This instance can be proven with `by infer_instance`. However, `WithBot ℕ` does not
pick up a `CovariantClass M M (Function.swap (*)) (≤)` instance without it (see PR mathlib#7940). -/
@[to_additive]
instance OrderedCommMonoid.toCovariantClassRight (M : Type*) [OrderedCommMonoid M] :
CovariantClass M M (swap (· * ·)) (· ≤ ·) :=
covariant_swap_mul_of_covariant_mul M _
#align ordered_comm_monoid.to_covariant_class_right OrderedCommMonoid.toCovariantClassRight
#align ordered_add_comm_monoid.to_covariant_class_right OrderedAddCommMonoid.toCovariantClassRight
end OrderedCommMonoid
/-- An ordered cancellative additive commutative monoid is a partially ordered commutative additive
monoid in which addition is cancellative and monotone. -/
class OrderedCancelAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α where
protected le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c
#align ordered_cancel_add_comm_monoid OrderedCancelAddCommMonoid
/-- An ordered cancellative commutative monoid is a partially ordered commutative monoid in which
multiplication is cancellative and monotone. -/
@[to_additive OrderedCancelAddCommMonoid]
class OrderedCancelCommMonoid (α : Type*) extends OrderedCommMonoid α where
protected le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c
#align ordered_cancel_comm_monoid OrderedCancelCommMonoid
#align ordered_cancel_comm_monoid.to_ordered_comm_monoid OrderedCancelCommMonoid.toOrderedCommMonoid
#align ordered_cancel_add_comm_monoid.to_ordered_add_comm_monoid OrderedCancelAddCommMonoid.toOrderedAddCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid α]
-- See note [lower instance priority]
@[to_additive]
instance (priority := 200) OrderedCancelCommMonoid.toContravariantClassLeLeft :
ContravariantClass α α (· * ·) (· ≤ ·) :=
⟨OrderedCancelCommMonoid.le_of_mul_le_mul_left⟩
#align ordered_cancel_comm_monoid.to_contravariant_class_le_left OrderedCancelCommMonoid.toContravariantClassLeLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_le_left OrderedCancelAddCommMonoid.toContravariantClassLeLeft
#noalign ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left
#noalign ordered_cancel_add_comm_monoid.lt_of_add_lt_add_left
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassLeft :
ContravariantClass α α (· * ·) (· < ·) where
elim := contravariant_lt_of_contravariant_le α α _ ContravariantClass.elim
#align ordered_cancel_comm_monoid.to_contravariant_class_left OrderedCancelCommMonoid.toContravariantClassLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_left OrderedCancelAddCommMonoid.toContravariantClassLeft
/- This instance can be proven with `by infer_instance`. However, by analogy with the
instance `OrderedCancelCommMonoid.to_covariantClass_right` above, I imagine that without
this instance, some Type would not have a `ContravariantClass M M (function.swap (*)) (<)`
instance. -/
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassRight :
ContravariantClass α α (swap (· * ·)) (· < ·) :=
contravariant_swap_mul_of_contravariant_mul α _
#align ordered_cancel_comm_monoid.to_contravariant_class_right OrderedCancelCommMonoid.toContravariantClassRight
#align ordered_cancel_add_comm_monoid.to_contravariant_class_right OrderedCancelAddCommMonoid.toContravariantClassRight
-- See note [lower instance priority]
@[to_additive OrderedCancelAddCommMonoid.toCancelAddCommMonoid]
instance (priority := 100) OrderedCancelCommMonoid.toCancelCommMonoid : CancelCommMonoid α :=
{ ‹OrderedCancelCommMonoid α› with
mul_left_cancel :=
fun a b c h => (le_of_mul_le_mul_left' h.le).antisymm <| le_of_mul_le_mul_left' h.ge }
#align ordered_cancel_comm_monoid.to_cancel_comm_monoid OrderedCancelCommMonoid.toCancelCommMonoid
#align ordered_cancel_add_comm_monoid.to_cancel_add_comm_monoid OrderedCancelAddCommMonoid.toCancelAddCommMonoid
#noalign has_mul.to_covariant_class_left
#noalign has_add.to_covariant_class_left
#noalign has_mul.to_covariant_class_right
#noalign has_add.to_covariant_class_right
end OrderedCancelCommMonoid
set_option linter.deprecated false in
@[deprecated] theorem bit0_pos [OrderedAddCommMonoid α] {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos' h h
#align bit0_pos bit0_pos
/-- A linearly ordered additive commutative monoid. -/
class LinearOrderedAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α, LinearOrder α
#align linear_ordered_add_comm_monoid LinearOrderedAddCommMonoid
/-- A linearly ordered commutative monoid. -/
@[to_additive]
class LinearOrderedCommMonoid (α : Type*) extends OrderedCommMonoid α, LinearOrder α
#align linear_ordered_comm_monoid LinearOrderedCommMonoid
/-- A linearly ordered cancellative additive commutative monoid is an additive commutative monoid
with a decidable linear order in which addition is cancellative and monotone. -/
class LinearOrderedCancelAddCommMonoid (α : Type*) extends OrderedCancelAddCommMonoid α,
LinearOrderedAddCommMonoid α
#align linear_ordered_cancel_add_comm_monoid LinearOrderedCancelAddCommMonoid
/-- A linearly ordered cancellative commutative monoid is a commutative monoid with a linear order
in which multiplication is cancellative and monotone. -/
@[to_additive LinearOrderedCancelAddCommMonoid]
class LinearOrderedCancelCommMonoid (α : Type*) extends OrderedCancelCommMonoid α,
LinearOrderedCommMonoid α
#align linear_ordered_cancel_comm_monoid LinearOrderedCancelCommMonoid
attribute [to_additive existing] LinearOrderedCancelCommMonoid.toLinearOrderedCommMonoid
/-- A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined. -/
class LinearOrderedAddCommMonoidWithTop (α : Type*) extends LinearOrderedAddCommMonoid α,
OrderTop α where
/-- In a `LinearOrderedAddCommMonoidWithTop`, the `⊤` element is invariant under addition. -/
protected top_add' : ∀ x : α, ⊤ + x = ⊤
#align linear_ordered_add_comm_monoid_with_top LinearOrderedAddCommMonoidWithTop
#align linear_ordered_add_comm_monoid_with_top.to_order_top LinearOrderedAddCommMonoidWithTop.toOrderTop
section LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedAddCommMonoidWithTop α] {a b : α}
@[simp]
theorem top_add (a : α) : ⊤ + a = ⊤ :=
LinearOrderedAddCommMonoidWithTop.top_add' a
#align top_add top_add
@[simp]
theorem add_top (a : α) : a + ⊤ = ⊤ :=
Trans.trans (add_comm _ _) (top_add _)
#align add_top add_top
end LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedCommMonoid α] {a : α}
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by
|
push_neg at h ⊢
|
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by
|
Mathlib.Algebra.Order.Monoid.Defs.176_0.P9s7TI8QJTWokXI
|
@[to_additive (attr
|
Mathlib_Algebra_Order_Monoid_Defs
|
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
h : a < 1
⊢ a * a < 1
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Ordered monoids
This file provides the definitions of ordered monoids.
-/
open Function
variable {α β : Type*}
/-- An ordered (additive) commutative monoid is a commutative monoid with a partial order such that
addition is monotone. -/
class OrderedAddCommMonoid (α : Type*) extends AddCommMonoid α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c, c + a ≤ c + b
#align ordered_add_comm_monoid OrderedAddCommMonoid
/-- An ordered commutative monoid is a commutative monoid with a partial order such that
multiplication is monotone. -/
@[to_additive]
class OrderedCommMonoid (α : Type*) extends CommMonoid α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c, c * a ≤ c * b
#align ordered_comm_monoid OrderedCommMonoid
section OrderedCommMonoid
variable [OrderedCommMonoid α]
@[to_additive]
instance OrderedCommMonoid.toCovariantClassLeft : CovariantClass α α (· * ·) (· ≤ ·) where
elim := fun a _ _ bc ↦ OrderedCommMonoid.mul_le_mul_left _ _ bc a
#align ordered_comm_monoid.to_covariant_class_left OrderedCommMonoid.toCovariantClassLeft
#align ordered_add_comm_monoid.to_covariant_class_left OrderedAddCommMonoid.toCovariantClassLeft
/- This instance can be proven with `by infer_instance`. However, `WithBot ℕ` does not
pick up a `CovariantClass M M (Function.swap (*)) (≤)` instance without it (see PR mathlib#7940). -/
@[to_additive]
instance OrderedCommMonoid.toCovariantClassRight (M : Type*) [OrderedCommMonoid M] :
CovariantClass M M (swap (· * ·)) (· ≤ ·) :=
covariant_swap_mul_of_covariant_mul M _
#align ordered_comm_monoid.to_covariant_class_right OrderedCommMonoid.toCovariantClassRight
#align ordered_add_comm_monoid.to_covariant_class_right OrderedAddCommMonoid.toCovariantClassRight
end OrderedCommMonoid
/-- An ordered cancellative additive commutative monoid is a partially ordered commutative additive
monoid in which addition is cancellative and monotone. -/
class OrderedCancelAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α where
protected le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c
#align ordered_cancel_add_comm_monoid OrderedCancelAddCommMonoid
/-- An ordered cancellative commutative monoid is a partially ordered commutative monoid in which
multiplication is cancellative and monotone. -/
@[to_additive OrderedCancelAddCommMonoid]
class OrderedCancelCommMonoid (α : Type*) extends OrderedCommMonoid α where
protected le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c
#align ordered_cancel_comm_monoid OrderedCancelCommMonoid
#align ordered_cancel_comm_monoid.to_ordered_comm_monoid OrderedCancelCommMonoid.toOrderedCommMonoid
#align ordered_cancel_add_comm_monoid.to_ordered_add_comm_monoid OrderedCancelAddCommMonoid.toOrderedAddCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid α]
-- See note [lower instance priority]
@[to_additive]
instance (priority := 200) OrderedCancelCommMonoid.toContravariantClassLeLeft :
ContravariantClass α α (· * ·) (· ≤ ·) :=
⟨OrderedCancelCommMonoid.le_of_mul_le_mul_left⟩
#align ordered_cancel_comm_monoid.to_contravariant_class_le_left OrderedCancelCommMonoid.toContravariantClassLeLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_le_left OrderedCancelAddCommMonoid.toContravariantClassLeLeft
#noalign ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left
#noalign ordered_cancel_add_comm_monoid.lt_of_add_lt_add_left
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassLeft :
ContravariantClass α α (· * ·) (· < ·) where
elim := contravariant_lt_of_contravariant_le α α _ ContravariantClass.elim
#align ordered_cancel_comm_monoid.to_contravariant_class_left OrderedCancelCommMonoid.toContravariantClassLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_left OrderedCancelAddCommMonoid.toContravariantClassLeft
/- This instance can be proven with `by infer_instance`. However, by analogy with the
instance `OrderedCancelCommMonoid.to_covariantClass_right` above, I imagine that without
this instance, some Type would not have a `ContravariantClass M M (function.swap (*)) (<)`
instance. -/
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassRight :
ContravariantClass α α (swap (· * ·)) (· < ·) :=
contravariant_swap_mul_of_contravariant_mul α _
#align ordered_cancel_comm_monoid.to_contravariant_class_right OrderedCancelCommMonoid.toContravariantClassRight
#align ordered_cancel_add_comm_monoid.to_contravariant_class_right OrderedCancelAddCommMonoid.toContravariantClassRight
-- See note [lower instance priority]
@[to_additive OrderedCancelAddCommMonoid.toCancelAddCommMonoid]
instance (priority := 100) OrderedCancelCommMonoid.toCancelCommMonoid : CancelCommMonoid α :=
{ ‹OrderedCancelCommMonoid α› with
mul_left_cancel :=
fun a b c h => (le_of_mul_le_mul_left' h.le).antisymm <| le_of_mul_le_mul_left' h.ge }
#align ordered_cancel_comm_monoid.to_cancel_comm_monoid OrderedCancelCommMonoid.toCancelCommMonoid
#align ordered_cancel_add_comm_monoid.to_cancel_add_comm_monoid OrderedCancelAddCommMonoid.toCancelAddCommMonoid
#noalign has_mul.to_covariant_class_left
#noalign has_add.to_covariant_class_left
#noalign has_mul.to_covariant_class_right
#noalign has_add.to_covariant_class_right
end OrderedCancelCommMonoid
set_option linter.deprecated false in
@[deprecated] theorem bit0_pos [OrderedAddCommMonoid α] {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos' h h
#align bit0_pos bit0_pos
/-- A linearly ordered additive commutative monoid. -/
class LinearOrderedAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α, LinearOrder α
#align linear_ordered_add_comm_monoid LinearOrderedAddCommMonoid
/-- A linearly ordered commutative monoid. -/
@[to_additive]
class LinearOrderedCommMonoid (α : Type*) extends OrderedCommMonoid α, LinearOrder α
#align linear_ordered_comm_monoid LinearOrderedCommMonoid
/-- A linearly ordered cancellative additive commutative monoid is an additive commutative monoid
with a decidable linear order in which addition is cancellative and monotone. -/
class LinearOrderedCancelAddCommMonoid (α : Type*) extends OrderedCancelAddCommMonoid α,
LinearOrderedAddCommMonoid α
#align linear_ordered_cancel_add_comm_monoid LinearOrderedCancelAddCommMonoid
/-- A linearly ordered cancellative commutative monoid is a commutative monoid with a linear order
in which multiplication is cancellative and monotone. -/
@[to_additive LinearOrderedCancelAddCommMonoid]
class LinearOrderedCancelCommMonoid (α : Type*) extends OrderedCancelCommMonoid α,
LinearOrderedCommMonoid α
#align linear_ordered_cancel_comm_monoid LinearOrderedCancelCommMonoid
attribute [to_additive existing] LinearOrderedCancelCommMonoid.toLinearOrderedCommMonoid
/-- A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined. -/
class LinearOrderedAddCommMonoidWithTop (α : Type*) extends LinearOrderedAddCommMonoid α,
OrderTop α where
/-- In a `LinearOrderedAddCommMonoidWithTop`, the `⊤` element is invariant under addition. -/
protected top_add' : ∀ x : α, ⊤ + x = ⊤
#align linear_ordered_add_comm_monoid_with_top LinearOrderedAddCommMonoidWithTop
#align linear_ordered_add_comm_monoid_with_top.to_order_top LinearOrderedAddCommMonoidWithTop.toOrderTop
section LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedAddCommMonoidWithTop α] {a b : α}
@[simp]
theorem top_add (a : α) : ⊤ + a = ⊤ :=
LinearOrderedAddCommMonoidWithTop.top_add' a
#align top_add top_add
@[simp]
theorem add_top (a : α) : a + ⊤ = ⊤ :=
Trans.trans (add_comm _ _) (top_add _)
#align add_top add_top
end LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedCommMonoid α] {a : α}
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by push_neg at h ⊢;
|
exact mul_lt_one' h h
|
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by push_neg at h ⊢;
|
Mathlib.Algebra.Order.Monoid.Defs.176_0.P9s7TI8QJTWokXI
|
@[to_additive (attr
|
Mathlib_Algebra_Order_Monoid_Defs
|
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
h : ¬1 < a
⊢ ¬1 < a * a
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Ordered monoids
This file provides the definitions of ordered monoids.
-/
open Function
variable {α β : Type*}
/-- An ordered (additive) commutative monoid is a commutative monoid with a partial order such that
addition is monotone. -/
class OrderedAddCommMonoid (α : Type*) extends AddCommMonoid α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c, c + a ≤ c + b
#align ordered_add_comm_monoid OrderedAddCommMonoid
/-- An ordered commutative monoid is a commutative monoid with a partial order such that
multiplication is monotone. -/
@[to_additive]
class OrderedCommMonoid (α : Type*) extends CommMonoid α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c, c * a ≤ c * b
#align ordered_comm_monoid OrderedCommMonoid
section OrderedCommMonoid
variable [OrderedCommMonoid α]
@[to_additive]
instance OrderedCommMonoid.toCovariantClassLeft : CovariantClass α α (· * ·) (· ≤ ·) where
elim := fun a _ _ bc ↦ OrderedCommMonoid.mul_le_mul_left _ _ bc a
#align ordered_comm_monoid.to_covariant_class_left OrderedCommMonoid.toCovariantClassLeft
#align ordered_add_comm_monoid.to_covariant_class_left OrderedAddCommMonoid.toCovariantClassLeft
/- This instance can be proven with `by infer_instance`. However, `WithBot ℕ` does not
pick up a `CovariantClass M M (Function.swap (*)) (≤)` instance without it (see PR mathlib#7940). -/
@[to_additive]
instance OrderedCommMonoid.toCovariantClassRight (M : Type*) [OrderedCommMonoid M] :
CovariantClass M M (swap (· * ·)) (· ≤ ·) :=
covariant_swap_mul_of_covariant_mul M _
#align ordered_comm_monoid.to_covariant_class_right OrderedCommMonoid.toCovariantClassRight
#align ordered_add_comm_monoid.to_covariant_class_right OrderedAddCommMonoid.toCovariantClassRight
end OrderedCommMonoid
/-- An ordered cancellative additive commutative monoid is a partially ordered commutative additive
monoid in which addition is cancellative and monotone. -/
class OrderedCancelAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α where
protected le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c
#align ordered_cancel_add_comm_monoid OrderedCancelAddCommMonoid
/-- An ordered cancellative commutative monoid is a partially ordered commutative monoid in which
multiplication is cancellative and monotone. -/
@[to_additive OrderedCancelAddCommMonoid]
class OrderedCancelCommMonoid (α : Type*) extends OrderedCommMonoid α where
protected le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c
#align ordered_cancel_comm_monoid OrderedCancelCommMonoid
#align ordered_cancel_comm_monoid.to_ordered_comm_monoid OrderedCancelCommMonoid.toOrderedCommMonoid
#align ordered_cancel_add_comm_monoid.to_ordered_add_comm_monoid OrderedCancelAddCommMonoid.toOrderedAddCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid α]
-- See note [lower instance priority]
@[to_additive]
instance (priority := 200) OrderedCancelCommMonoid.toContravariantClassLeLeft :
ContravariantClass α α (· * ·) (· ≤ ·) :=
⟨OrderedCancelCommMonoid.le_of_mul_le_mul_left⟩
#align ordered_cancel_comm_monoid.to_contravariant_class_le_left OrderedCancelCommMonoid.toContravariantClassLeLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_le_left OrderedCancelAddCommMonoid.toContravariantClassLeLeft
#noalign ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left
#noalign ordered_cancel_add_comm_monoid.lt_of_add_lt_add_left
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassLeft :
ContravariantClass α α (· * ·) (· < ·) where
elim := contravariant_lt_of_contravariant_le α α _ ContravariantClass.elim
#align ordered_cancel_comm_monoid.to_contravariant_class_left OrderedCancelCommMonoid.toContravariantClassLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_left OrderedCancelAddCommMonoid.toContravariantClassLeft
/- This instance can be proven with `by infer_instance`. However, by analogy with the
instance `OrderedCancelCommMonoid.to_covariantClass_right` above, I imagine that without
this instance, some Type would not have a `ContravariantClass M M (function.swap (*)) (<)`
instance. -/
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassRight :
ContravariantClass α α (swap (· * ·)) (· < ·) :=
contravariant_swap_mul_of_contravariant_mul α _
#align ordered_cancel_comm_monoid.to_contravariant_class_right OrderedCancelCommMonoid.toContravariantClassRight
#align ordered_cancel_add_comm_monoid.to_contravariant_class_right OrderedCancelAddCommMonoid.toContravariantClassRight
-- See note [lower instance priority]
@[to_additive OrderedCancelAddCommMonoid.toCancelAddCommMonoid]
instance (priority := 100) OrderedCancelCommMonoid.toCancelCommMonoid : CancelCommMonoid α :=
{ ‹OrderedCancelCommMonoid α› with
mul_left_cancel :=
fun a b c h => (le_of_mul_le_mul_left' h.le).antisymm <| le_of_mul_le_mul_left' h.ge }
#align ordered_cancel_comm_monoid.to_cancel_comm_monoid OrderedCancelCommMonoid.toCancelCommMonoid
#align ordered_cancel_add_comm_monoid.to_cancel_add_comm_monoid OrderedCancelAddCommMonoid.toCancelAddCommMonoid
#noalign has_mul.to_covariant_class_left
#noalign has_add.to_covariant_class_left
#noalign has_mul.to_covariant_class_right
#noalign has_add.to_covariant_class_right
end OrderedCancelCommMonoid
set_option linter.deprecated false in
@[deprecated] theorem bit0_pos [OrderedAddCommMonoid α] {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos' h h
#align bit0_pos bit0_pos
/-- A linearly ordered additive commutative monoid. -/
class LinearOrderedAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α, LinearOrder α
#align linear_ordered_add_comm_monoid LinearOrderedAddCommMonoid
/-- A linearly ordered commutative monoid. -/
@[to_additive]
class LinearOrderedCommMonoid (α : Type*) extends OrderedCommMonoid α, LinearOrder α
#align linear_ordered_comm_monoid LinearOrderedCommMonoid
/-- A linearly ordered cancellative additive commutative monoid is an additive commutative monoid
with a decidable linear order in which addition is cancellative and monotone. -/
class LinearOrderedCancelAddCommMonoid (α : Type*) extends OrderedCancelAddCommMonoid α,
LinearOrderedAddCommMonoid α
#align linear_ordered_cancel_add_comm_monoid LinearOrderedCancelAddCommMonoid
/-- A linearly ordered cancellative commutative monoid is a commutative monoid with a linear order
in which multiplication is cancellative and monotone. -/
@[to_additive LinearOrderedCancelAddCommMonoid]
class LinearOrderedCancelCommMonoid (α : Type*) extends OrderedCancelCommMonoid α,
LinearOrderedCommMonoid α
#align linear_ordered_cancel_comm_monoid LinearOrderedCancelCommMonoid
attribute [to_additive existing] LinearOrderedCancelCommMonoid.toLinearOrderedCommMonoid
/-- A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined. -/
class LinearOrderedAddCommMonoidWithTop (α : Type*) extends LinearOrderedAddCommMonoid α,
OrderTop α where
/-- In a `LinearOrderedAddCommMonoidWithTop`, the `⊤` element is invariant under addition. -/
protected top_add' : ∀ x : α, ⊤ + x = ⊤
#align linear_ordered_add_comm_monoid_with_top LinearOrderedAddCommMonoidWithTop
#align linear_ordered_add_comm_monoid_with_top.to_order_top LinearOrderedAddCommMonoidWithTop.toOrderTop
section LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedAddCommMonoidWithTop α] {a b : α}
@[simp]
theorem top_add (a : α) : ⊤ + a = ⊤ :=
LinearOrderedAddCommMonoidWithTop.top_add' a
#align top_add top_add
@[simp]
theorem add_top (a : α) : a + ⊤ = ⊤ :=
Trans.trans (add_comm _ _) (top_add _)
#align add_top add_top
end LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedCommMonoid α] {a : α}
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by push_neg at h ⊢; exact mul_lt_one' h h).mtr, fun h ↦ one_le_mul h h⟩
@[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
⟨(fun h ↦ by
|
push_neg at h ⊢
|
@[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
⟨(fun h ↦ by
|
Mathlib.Algebra.Order.Monoid.Defs.180_0.P9s7TI8QJTWokXI
|
@[to_additive (attr
|
Mathlib_Algebra_Order_Monoid_Defs
|
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
h : a ≤ 1
⊢ a * a ≤ 1
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Ordered monoids
This file provides the definitions of ordered monoids.
-/
open Function
variable {α β : Type*}
/-- An ordered (additive) commutative monoid is a commutative monoid with a partial order such that
addition is monotone. -/
class OrderedAddCommMonoid (α : Type*) extends AddCommMonoid α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c, c + a ≤ c + b
#align ordered_add_comm_monoid OrderedAddCommMonoid
/-- An ordered commutative monoid is a commutative monoid with a partial order such that
multiplication is monotone. -/
@[to_additive]
class OrderedCommMonoid (α : Type*) extends CommMonoid α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c, c * a ≤ c * b
#align ordered_comm_monoid OrderedCommMonoid
section OrderedCommMonoid
variable [OrderedCommMonoid α]
@[to_additive]
instance OrderedCommMonoid.toCovariantClassLeft : CovariantClass α α (· * ·) (· ≤ ·) where
elim := fun a _ _ bc ↦ OrderedCommMonoid.mul_le_mul_left _ _ bc a
#align ordered_comm_monoid.to_covariant_class_left OrderedCommMonoid.toCovariantClassLeft
#align ordered_add_comm_monoid.to_covariant_class_left OrderedAddCommMonoid.toCovariantClassLeft
/- This instance can be proven with `by infer_instance`. However, `WithBot ℕ` does not
pick up a `CovariantClass M M (Function.swap (*)) (≤)` instance without it (see PR mathlib#7940). -/
@[to_additive]
instance OrderedCommMonoid.toCovariantClassRight (M : Type*) [OrderedCommMonoid M] :
CovariantClass M M (swap (· * ·)) (· ≤ ·) :=
covariant_swap_mul_of_covariant_mul M _
#align ordered_comm_monoid.to_covariant_class_right OrderedCommMonoid.toCovariantClassRight
#align ordered_add_comm_monoid.to_covariant_class_right OrderedAddCommMonoid.toCovariantClassRight
end OrderedCommMonoid
/-- An ordered cancellative additive commutative monoid is a partially ordered commutative additive
monoid in which addition is cancellative and monotone. -/
class OrderedCancelAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α where
protected le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c
#align ordered_cancel_add_comm_monoid OrderedCancelAddCommMonoid
/-- An ordered cancellative commutative monoid is a partially ordered commutative monoid in which
multiplication is cancellative and monotone. -/
@[to_additive OrderedCancelAddCommMonoid]
class OrderedCancelCommMonoid (α : Type*) extends OrderedCommMonoid α where
protected le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c
#align ordered_cancel_comm_monoid OrderedCancelCommMonoid
#align ordered_cancel_comm_monoid.to_ordered_comm_monoid OrderedCancelCommMonoid.toOrderedCommMonoid
#align ordered_cancel_add_comm_monoid.to_ordered_add_comm_monoid OrderedCancelAddCommMonoid.toOrderedAddCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid α]
-- See note [lower instance priority]
@[to_additive]
instance (priority := 200) OrderedCancelCommMonoid.toContravariantClassLeLeft :
ContravariantClass α α (· * ·) (· ≤ ·) :=
⟨OrderedCancelCommMonoid.le_of_mul_le_mul_left⟩
#align ordered_cancel_comm_monoid.to_contravariant_class_le_left OrderedCancelCommMonoid.toContravariantClassLeLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_le_left OrderedCancelAddCommMonoid.toContravariantClassLeLeft
#noalign ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left
#noalign ordered_cancel_add_comm_monoid.lt_of_add_lt_add_left
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassLeft :
ContravariantClass α α (· * ·) (· < ·) where
elim := contravariant_lt_of_contravariant_le α α _ ContravariantClass.elim
#align ordered_cancel_comm_monoid.to_contravariant_class_left OrderedCancelCommMonoid.toContravariantClassLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_left OrderedCancelAddCommMonoid.toContravariantClassLeft
/- This instance can be proven with `by infer_instance`. However, by analogy with the
instance `OrderedCancelCommMonoid.to_covariantClass_right` above, I imagine that without
this instance, some Type would not have a `ContravariantClass M M (function.swap (*)) (<)`
instance. -/
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassRight :
ContravariantClass α α (swap (· * ·)) (· < ·) :=
contravariant_swap_mul_of_contravariant_mul α _
#align ordered_cancel_comm_monoid.to_contravariant_class_right OrderedCancelCommMonoid.toContravariantClassRight
#align ordered_cancel_add_comm_monoid.to_contravariant_class_right OrderedCancelAddCommMonoid.toContravariantClassRight
-- See note [lower instance priority]
@[to_additive OrderedCancelAddCommMonoid.toCancelAddCommMonoid]
instance (priority := 100) OrderedCancelCommMonoid.toCancelCommMonoid : CancelCommMonoid α :=
{ ‹OrderedCancelCommMonoid α› with
mul_left_cancel :=
fun a b c h => (le_of_mul_le_mul_left' h.le).antisymm <| le_of_mul_le_mul_left' h.ge }
#align ordered_cancel_comm_monoid.to_cancel_comm_monoid OrderedCancelCommMonoid.toCancelCommMonoid
#align ordered_cancel_add_comm_monoid.to_cancel_add_comm_monoid OrderedCancelAddCommMonoid.toCancelAddCommMonoid
#noalign has_mul.to_covariant_class_left
#noalign has_add.to_covariant_class_left
#noalign has_mul.to_covariant_class_right
#noalign has_add.to_covariant_class_right
end OrderedCancelCommMonoid
set_option linter.deprecated false in
@[deprecated] theorem bit0_pos [OrderedAddCommMonoid α] {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos' h h
#align bit0_pos bit0_pos
/-- A linearly ordered additive commutative monoid. -/
class LinearOrderedAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α, LinearOrder α
#align linear_ordered_add_comm_monoid LinearOrderedAddCommMonoid
/-- A linearly ordered commutative monoid. -/
@[to_additive]
class LinearOrderedCommMonoid (α : Type*) extends OrderedCommMonoid α, LinearOrder α
#align linear_ordered_comm_monoid LinearOrderedCommMonoid
/-- A linearly ordered cancellative additive commutative monoid is an additive commutative monoid
with a decidable linear order in which addition is cancellative and monotone. -/
class LinearOrderedCancelAddCommMonoid (α : Type*) extends OrderedCancelAddCommMonoid α,
LinearOrderedAddCommMonoid α
#align linear_ordered_cancel_add_comm_monoid LinearOrderedCancelAddCommMonoid
/-- A linearly ordered cancellative commutative monoid is a commutative monoid with a linear order
in which multiplication is cancellative and monotone. -/
@[to_additive LinearOrderedCancelAddCommMonoid]
class LinearOrderedCancelCommMonoid (α : Type*) extends OrderedCancelCommMonoid α,
LinearOrderedCommMonoid α
#align linear_ordered_cancel_comm_monoid LinearOrderedCancelCommMonoid
attribute [to_additive existing] LinearOrderedCancelCommMonoid.toLinearOrderedCommMonoid
/-- A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined. -/
class LinearOrderedAddCommMonoidWithTop (α : Type*) extends LinearOrderedAddCommMonoid α,
OrderTop α where
/-- In a `LinearOrderedAddCommMonoidWithTop`, the `⊤` element is invariant under addition. -/
protected top_add' : ∀ x : α, ⊤ + x = ⊤
#align linear_ordered_add_comm_monoid_with_top LinearOrderedAddCommMonoidWithTop
#align linear_ordered_add_comm_monoid_with_top.to_order_top LinearOrderedAddCommMonoidWithTop.toOrderTop
section LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedAddCommMonoidWithTop α] {a b : α}
@[simp]
theorem top_add (a : α) : ⊤ + a = ⊤ :=
LinearOrderedAddCommMonoidWithTop.top_add' a
#align top_add top_add
@[simp]
theorem add_top (a : α) : a + ⊤ = ⊤ :=
Trans.trans (add_comm _ _) (top_add _)
#align add_top add_top
end LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedCommMonoid α] {a : α}
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by push_neg at h ⊢; exact mul_lt_one' h h).mtr, fun h ↦ one_le_mul h h⟩
@[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
⟨(fun h ↦ by push_neg at h ⊢;
|
exact mul_le_one' h h
|
@[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
⟨(fun h ↦ by push_neg at h ⊢;
|
Mathlib.Algebra.Order.Monoid.Defs.180_0.P9s7TI8QJTWokXI
|
@[to_additive (attr
|
Mathlib_Algebra_Order_Monoid_Defs
|
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
⊢ a * a ≤ 1 ↔ a ≤ 1
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Ordered monoids
This file provides the definitions of ordered monoids.
-/
open Function
variable {α β : Type*}
/-- An ordered (additive) commutative monoid is a commutative monoid with a partial order such that
addition is monotone. -/
class OrderedAddCommMonoid (α : Type*) extends AddCommMonoid α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c, c + a ≤ c + b
#align ordered_add_comm_monoid OrderedAddCommMonoid
/-- An ordered commutative monoid is a commutative monoid with a partial order such that
multiplication is monotone. -/
@[to_additive]
class OrderedCommMonoid (α : Type*) extends CommMonoid α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c, c * a ≤ c * b
#align ordered_comm_monoid OrderedCommMonoid
section OrderedCommMonoid
variable [OrderedCommMonoid α]
@[to_additive]
instance OrderedCommMonoid.toCovariantClassLeft : CovariantClass α α (· * ·) (· ≤ ·) where
elim := fun a _ _ bc ↦ OrderedCommMonoid.mul_le_mul_left _ _ bc a
#align ordered_comm_monoid.to_covariant_class_left OrderedCommMonoid.toCovariantClassLeft
#align ordered_add_comm_monoid.to_covariant_class_left OrderedAddCommMonoid.toCovariantClassLeft
/- This instance can be proven with `by infer_instance`. However, `WithBot ℕ` does not
pick up a `CovariantClass M M (Function.swap (*)) (≤)` instance without it (see PR mathlib#7940). -/
@[to_additive]
instance OrderedCommMonoid.toCovariantClassRight (M : Type*) [OrderedCommMonoid M] :
CovariantClass M M (swap (· * ·)) (· ≤ ·) :=
covariant_swap_mul_of_covariant_mul M _
#align ordered_comm_monoid.to_covariant_class_right OrderedCommMonoid.toCovariantClassRight
#align ordered_add_comm_monoid.to_covariant_class_right OrderedAddCommMonoid.toCovariantClassRight
end OrderedCommMonoid
/-- An ordered cancellative additive commutative monoid is a partially ordered commutative additive
monoid in which addition is cancellative and monotone. -/
class OrderedCancelAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α where
protected le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c
#align ordered_cancel_add_comm_monoid OrderedCancelAddCommMonoid
/-- An ordered cancellative commutative monoid is a partially ordered commutative monoid in which
multiplication is cancellative and monotone. -/
@[to_additive OrderedCancelAddCommMonoid]
class OrderedCancelCommMonoid (α : Type*) extends OrderedCommMonoid α where
protected le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c
#align ordered_cancel_comm_monoid OrderedCancelCommMonoid
#align ordered_cancel_comm_monoid.to_ordered_comm_monoid OrderedCancelCommMonoid.toOrderedCommMonoid
#align ordered_cancel_add_comm_monoid.to_ordered_add_comm_monoid OrderedCancelAddCommMonoid.toOrderedAddCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid α]
-- See note [lower instance priority]
@[to_additive]
instance (priority := 200) OrderedCancelCommMonoid.toContravariantClassLeLeft :
ContravariantClass α α (· * ·) (· ≤ ·) :=
⟨OrderedCancelCommMonoid.le_of_mul_le_mul_left⟩
#align ordered_cancel_comm_monoid.to_contravariant_class_le_left OrderedCancelCommMonoid.toContravariantClassLeLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_le_left OrderedCancelAddCommMonoid.toContravariantClassLeLeft
#noalign ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left
#noalign ordered_cancel_add_comm_monoid.lt_of_add_lt_add_left
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassLeft :
ContravariantClass α α (· * ·) (· < ·) where
elim := contravariant_lt_of_contravariant_le α α _ ContravariantClass.elim
#align ordered_cancel_comm_monoid.to_contravariant_class_left OrderedCancelCommMonoid.toContravariantClassLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_left OrderedCancelAddCommMonoid.toContravariantClassLeft
/- This instance can be proven with `by infer_instance`. However, by analogy with the
instance `OrderedCancelCommMonoid.to_covariantClass_right` above, I imagine that without
this instance, some Type would not have a `ContravariantClass M M (function.swap (*)) (<)`
instance. -/
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassRight :
ContravariantClass α α (swap (· * ·)) (· < ·) :=
contravariant_swap_mul_of_contravariant_mul α _
#align ordered_cancel_comm_monoid.to_contravariant_class_right OrderedCancelCommMonoid.toContravariantClassRight
#align ordered_cancel_add_comm_monoid.to_contravariant_class_right OrderedCancelAddCommMonoid.toContravariantClassRight
-- See note [lower instance priority]
@[to_additive OrderedCancelAddCommMonoid.toCancelAddCommMonoid]
instance (priority := 100) OrderedCancelCommMonoid.toCancelCommMonoid : CancelCommMonoid α :=
{ ‹OrderedCancelCommMonoid α› with
mul_left_cancel :=
fun a b c h => (le_of_mul_le_mul_left' h.le).antisymm <| le_of_mul_le_mul_left' h.ge }
#align ordered_cancel_comm_monoid.to_cancel_comm_monoid OrderedCancelCommMonoid.toCancelCommMonoid
#align ordered_cancel_add_comm_monoid.to_cancel_add_comm_monoid OrderedCancelAddCommMonoid.toCancelAddCommMonoid
#noalign has_mul.to_covariant_class_left
#noalign has_add.to_covariant_class_left
#noalign has_mul.to_covariant_class_right
#noalign has_add.to_covariant_class_right
end OrderedCancelCommMonoid
set_option linter.deprecated false in
@[deprecated] theorem bit0_pos [OrderedAddCommMonoid α] {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos' h h
#align bit0_pos bit0_pos
/-- A linearly ordered additive commutative monoid. -/
class LinearOrderedAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α, LinearOrder α
#align linear_ordered_add_comm_monoid LinearOrderedAddCommMonoid
/-- A linearly ordered commutative monoid. -/
@[to_additive]
class LinearOrderedCommMonoid (α : Type*) extends OrderedCommMonoid α, LinearOrder α
#align linear_ordered_comm_monoid LinearOrderedCommMonoid
/-- A linearly ordered cancellative additive commutative monoid is an additive commutative monoid
with a decidable linear order in which addition is cancellative and monotone. -/
class LinearOrderedCancelAddCommMonoid (α : Type*) extends OrderedCancelAddCommMonoid α,
LinearOrderedAddCommMonoid α
#align linear_ordered_cancel_add_comm_monoid LinearOrderedCancelAddCommMonoid
/-- A linearly ordered cancellative commutative monoid is a commutative monoid with a linear order
in which multiplication is cancellative and monotone. -/
@[to_additive LinearOrderedCancelAddCommMonoid]
class LinearOrderedCancelCommMonoid (α : Type*) extends OrderedCancelCommMonoid α,
LinearOrderedCommMonoid α
#align linear_ordered_cancel_comm_monoid LinearOrderedCancelCommMonoid
attribute [to_additive existing] LinearOrderedCancelCommMonoid.toLinearOrderedCommMonoid
/-- A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined. -/
class LinearOrderedAddCommMonoidWithTop (α : Type*) extends LinearOrderedAddCommMonoid α,
OrderTop α where
/-- In a `LinearOrderedAddCommMonoidWithTop`, the `⊤` element is invariant under addition. -/
protected top_add' : ∀ x : α, ⊤ + x = ⊤
#align linear_ordered_add_comm_monoid_with_top LinearOrderedAddCommMonoidWithTop
#align linear_ordered_add_comm_monoid_with_top.to_order_top LinearOrderedAddCommMonoidWithTop.toOrderTop
section LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedAddCommMonoidWithTop α] {a b : α}
@[simp]
theorem top_add (a : α) : ⊤ + a = ⊤ :=
LinearOrderedAddCommMonoidWithTop.top_add' a
#align top_add top_add
@[simp]
theorem add_top (a : α) : a + ⊤ = ⊤ :=
Trans.trans (add_comm _ _) (top_add _)
#align add_top add_top
end LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedCommMonoid α] {a : α}
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by push_neg at h ⊢; exact mul_lt_one' h h).mtr, fun h ↦ one_le_mul h h⟩
@[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
⟨(fun h ↦ by push_neg at h ⊢; exact mul_le_one' h h).mtr, fun h ↦ one_lt_mul'' h h⟩
@[to_additive (attr := simp)]
theorem mul_self_le_one_iff : a * a ≤ 1 ↔ a ≤ 1 := by
|
simp [← not_iff_not]
|
@[to_additive (attr := simp)]
theorem mul_self_le_one_iff : a * a ≤ 1 ↔ a ≤ 1 := by
|
Mathlib.Algebra.Order.Monoid.Defs.184_0.P9s7TI8QJTWokXI
|
@[to_additive (attr
|
Mathlib_Algebra_Order_Monoid_Defs
|
α : Type u_1
β : Type u_2
inst✝ : LinearOrderedCommMonoid α
a : α
⊢ a * a < 1 ↔ a < 1
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Lemmas
import Mathlib.Order.BoundedOrder
#align_import algebra.order.monoid.defs from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# Ordered monoids
This file provides the definitions of ordered monoids.
-/
open Function
variable {α β : Type*}
/-- An ordered (additive) commutative monoid is a commutative monoid with a partial order such that
addition is monotone. -/
class OrderedAddCommMonoid (α : Type*) extends AddCommMonoid α, PartialOrder α where
protected add_le_add_left : ∀ a b : α, a ≤ b → ∀ c, c + a ≤ c + b
#align ordered_add_comm_monoid OrderedAddCommMonoid
/-- An ordered commutative monoid is a commutative monoid with a partial order such that
multiplication is monotone. -/
@[to_additive]
class OrderedCommMonoid (α : Type*) extends CommMonoid α, PartialOrder α where
protected mul_le_mul_left : ∀ a b : α, a ≤ b → ∀ c, c * a ≤ c * b
#align ordered_comm_monoid OrderedCommMonoid
section OrderedCommMonoid
variable [OrderedCommMonoid α]
@[to_additive]
instance OrderedCommMonoid.toCovariantClassLeft : CovariantClass α α (· * ·) (· ≤ ·) where
elim := fun a _ _ bc ↦ OrderedCommMonoid.mul_le_mul_left _ _ bc a
#align ordered_comm_monoid.to_covariant_class_left OrderedCommMonoid.toCovariantClassLeft
#align ordered_add_comm_monoid.to_covariant_class_left OrderedAddCommMonoid.toCovariantClassLeft
/- This instance can be proven with `by infer_instance`. However, `WithBot ℕ` does not
pick up a `CovariantClass M M (Function.swap (*)) (≤)` instance without it (see PR mathlib#7940). -/
@[to_additive]
instance OrderedCommMonoid.toCovariantClassRight (M : Type*) [OrderedCommMonoid M] :
CovariantClass M M (swap (· * ·)) (· ≤ ·) :=
covariant_swap_mul_of_covariant_mul M _
#align ordered_comm_monoid.to_covariant_class_right OrderedCommMonoid.toCovariantClassRight
#align ordered_add_comm_monoid.to_covariant_class_right OrderedAddCommMonoid.toCovariantClassRight
end OrderedCommMonoid
/-- An ordered cancellative additive commutative monoid is a partially ordered commutative additive
monoid in which addition is cancellative and monotone. -/
class OrderedCancelAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α where
protected le_of_add_le_add_left : ∀ a b c : α, a + b ≤ a + c → b ≤ c
#align ordered_cancel_add_comm_monoid OrderedCancelAddCommMonoid
/-- An ordered cancellative commutative monoid is a partially ordered commutative monoid in which
multiplication is cancellative and monotone. -/
@[to_additive OrderedCancelAddCommMonoid]
class OrderedCancelCommMonoid (α : Type*) extends OrderedCommMonoid α where
protected le_of_mul_le_mul_left : ∀ a b c : α, a * b ≤ a * c → b ≤ c
#align ordered_cancel_comm_monoid OrderedCancelCommMonoid
#align ordered_cancel_comm_monoid.to_ordered_comm_monoid OrderedCancelCommMonoid.toOrderedCommMonoid
#align ordered_cancel_add_comm_monoid.to_ordered_add_comm_monoid OrderedCancelAddCommMonoid.toOrderedAddCommMonoid
section OrderedCancelCommMonoid
variable [OrderedCancelCommMonoid α]
-- See note [lower instance priority]
@[to_additive]
instance (priority := 200) OrderedCancelCommMonoid.toContravariantClassLeLeft :
ContravariantClass α α (· * ·) (· ≤ ·) :=
⟨OrderedCancelCommMonoid.le_of_mul_le_mul_left⟩
#align ordered_cancel_comm_monoid.to_contravariant_class_le_left OrderedCancelCommMonoid.toContravariantClassLeLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_le_left OrderedCancelAddCommMonoid.toContravariantClassLeLeft
#noalign ordered_cancel_comm_monoid.lt_of_mul_lt_mul_left
#noalign ordered_cancel_add_comm_monoid.lt_of_add_lt_add_left
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassLeft :
ContravariantClass α α (· * ·) (· < ·) where
elim := contravariant_lt_of_contravariant_le α α _ ContravariantClass.elim
#align ordered_cancel_comm_monoid.to_contravariant_class_left OrderedCancelCommMonoid.toContravariantClassLeft
#align ordered_cancel_add_comm_monoid.to_contravariant_class_left OrderedCancelAddCommMonoid.toContravariantClassLeft
/- This instance can be proven with `by infer_instance`. However, by analogy with the
instance `OrderedCancelCommMonoid.to_covariantClass_right` above, I imagine that without
this instance, some Type would not have a `ContravariantClass M M (function.swap (*)) (<)`
instance. -/
@[to_additive]
instance OrderedCancelCommMonoid.toContravariantClassRight :
ContravariantClass α α (swap (· * ·)) (· < ·) :=
contravariant_swap_mul_of_contravariant_mul α _
#align ordered_cancel_comm_monoid.to_contravariant_class_right OrderedCancelCommMonoid.toContravariantClassRight
#align ordered_cancel_add_comm_monoid.to_contravariant_class_right OrderedCancelAddCommMonoid.toContravariantClassRight
-- See note [lower instance priority]
@[to_additive OrderedCancelAddCommMonoid.toCancelAddCommMonoid]
instance (priority := 100) OrderedCancelCommMonoid.toCancelCommMonoid : CancelCommMonoid α :=
{ ‹OrderedCancelCommMonoid α› with
mul_left_cancel :=
fun a b c h => (le_of_mul_le_mul_left' h.le).antisymm <| le_of_mul_le_mul_left' h.ge }
#align ordered_cancel_comm_monoid.to_cancel_comm_monoid OrderedCancelCommMonoid.toCancelCommMonoid
#align ordered_cancel_add_comm_monoid.to_cancel_add_comm_monoid OrderedCancelAddCommMonoid.toCancelAddCommMonoid
#noalign has_mul.to_covariant_class_left
#noalign has_add.to_covariant_class_left
#noalign has_mul.to_covariant_class_right
#noalign has_add.to_covariant_class_right
end OrderedCancelCommMonoid
set_option linter.deprecated false in
@[deprecated] theorem bit0_pos [OrderedAddCommMonoid α] {a : α} (h : 0 < a) : 0 < bit0 a :=
add_pos' h h
#align bit0_pos bit0_pos
/-- A linearly ordered additive commutative monoid. -/
class LinearOrderedAddCommMonoid (α : Type*) extends OrderedAddCommMonoid α, LinearOrder α
#align linear_ordered_add_comm_monoid LinearOrderedAddCommMonoid
/-- A linearly ordered commutative monoid. -/
@[to_additive]
class LinearOrderedCommMonoid (α : Type*) extends OrderedCommMonoid α, LinearOrder α
#align linear_ordered_comm_monoid LinearOrderedCommMonoid
/-- A linearly ordered cancellative additive commutative monoid is an additive commutative monoid
with a decidable linear order in which addition is cancellative and monotone. -/
class LinearOrderedCancelAddCommMonoid (α : Type*) extends OrderedCancelAddCommMonoid α,
LinearOrderedAddCommMonoid α
#align linear_ordered_cancel_add_comm_monoid LinearOrderedCancelAddCommMonoid
/-- A linearly ordered cancellative commutative monoid is a commutative monoid with a linear order
in which multiplication is cancellative and monotone. -/
@[to_additive LinearOrderedCancelAddCommMonoid]
class LinearOrderedCancelCommMonoid (α : Type*) extends OrderedCancelCommMonoid α,
LinearOrderedCommMonoid α
#align linear_ordered_cancel_comm_monoid LinearOrderedCancelCommMonoid
attribute [to_additive existing] LinearOrderedCancelCommMonoid.toLinearOrderedCommMonoid
/-- A linearly ordered commutative monoid with an additively absorbing `⊤` element.
Instances should include number systems with an infinite element adjoined. -/
class LinearOrderedAddCommMonoidWithTop (α : Type*) extends LinearOrderedAddCommMonoid α,
OrderTop α where
/-- In a `LinearOrderedAddCommMonoidWithTop`, the `⊤` element is invariant under addition. -/
protected top_add' : ∀ x : α, ⊤ + x = ⊤
#align linear_ordered_add_comm_monoid_with_top LinearOrderedAddCommMonoidWithTop
#align linear_ordered_add_comm_monoid_with_top.to_order_top LinearOrderedAddCommMonoidWithTop.toOrderTop
section LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedAddCommMonoidWithTop α] {a b : α}
@[simp]
theorem top_add (a : α) : ⊤ + a = ⊤ :=
LinearOrderedAddCommMonoidWithTop.top_add' a
#align top_add top_add
@[simp]
theorem add_top (a : α) : a + ⊤ = ⊤ :=
Trans.trans (add_comm _ _) (top_add _)
#align add_top add_top
end LinearOrderedAddCommMonoidWithTop
variable [LinearOrderedCommMonoid α] {a : α}
@[to_additive (attr := simp)]
theorem one_le_mul_self_iff : 1 ≤ a * a ↔ 1 ≤ a :=
⟨(fun h ↦ by push_neg at h ⊢; exact mul_lt_one' h h).mtr, fun h ↦ one_le_mul h h⟩
@[to_additive (attr := simp)]
theorem one_lt_mul_self_iff : 1 < a * a ↔ 1 < a :=
⟨(fun h ↦ by push_neg at h ⊢; exact mul_le_one' h h).mtr, fun h ↦ one_lt_mul'' h h⟩
@[to_additive (attr := simp)]
theorem mul_self_le_one_iff : a * a ≤ 1 ↔ a ≤ 1 := by simp [← not_iff_not]
@[to_additive (attr := simp)]
theorem mul_self_lt_one_iff : a * a < 1 ↔ a < 1 := by
|
simp [← not_iff_not]
|
@[to_additive (attr := simp)]
theorem mul_self_lt_one_iff : a * a < 1 ↔ a < 1 := by
|
Mathlib.Algebra.Order.Monoid.Defs.187_0.P9s7TI8QJTWokXI
|
@[to_additive (attr
|
Mathlib_Algebra_Order_Monoid_Defs
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : Setoid.r a₁ b₁
h₂ : Setoid.r a₂ b₂
⊢ Setoid.r (a₁ * a₂) (b₁ * b₂)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
|
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
|
Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : a₁ - b₁ ∈ I
h₂ : a₂ - b₂ ∈ I
⊢ a₁ * a₂ - b₁ * b₂ ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
|
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
|
Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : a₁ - b₁ ∈ I
h₂ : a₂ - b₂ ∈ I
F : a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ ∈ I
⊢ a₁ * a₂ - b₁ * b₂ ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
|
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
|
Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : a₁ - b₁ ∈ I
h₂ : a₂ - b₂ ∈ I
F : a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ ∈ I
⊢ a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
|
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
|
Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
src✝ : AddCon R := QuotientAddGroup.con (Submodule.toAddSubgroup I)
a₁ b₁ a₂ b₂ : R
h₁ : a₁ - b₁ ∈ I
h₂ : a₂ - b₂ ∈ I
F : a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ ∈ I
this : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁
⊢ a₁ * a₂ - b₁ * b₂ ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
|
rwa [← this] at F
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
|
Mathlib.RingTheory.Ideal.Quotient.61_0.TwNAv7Pc4PYOWjX
|
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S : Type v
x✝ y✝ x y : R
⊢ (mk (span {x})) y = 0 ↔ x ∣ y
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
|
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
|
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
|
Mathlib.RingTheory.Ideal.Quotient.133_0.TwNAv7Pc4PYOWjX
|
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S : Type v
x✝ y✝ x y : R
⊢ (mk I) x = (mk I) y ↔ x - y ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
|
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
|
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
|
Mathlib.RingTheory.Ideal.Quotient.137_0.TwNAv7Pc4PYOWjX
|
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
⊢ Subsingleton (R ⧸ I) ↔ I = ⊤
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
|
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
|
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
|
Mathlib.RingTheory.Ideal.Quotient.152_0.TwNAv7Pc4PYOWjX
|
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S : Type v
x y : R
⊢ ∀ (a : R ⧸ ⊤), a = default
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by
|
rintro ⟨x⟩
|
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by
|
Mathlib.RingTheory.Ideal.Quotient.157_0.TwNAv7Pc4PYOWjX
|
instance : Unique (R ⧸ (⊤ : Ideal R))
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S : Type v
x✝ y : R
a✝ : R ⧸ ⊤
x : R
⊢ Quot.mk Setoid.r x = default
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩;
|
exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top
|
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩;
|
Mathlib.RingTheory.Ideal.Quotient.157_0.TwNAv7Pc4PYOWjX
|
instance : Unique (R ⧸ (⊤ : Ideal R))
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
s : Set R
⊢ ⇑(mk I) ⁻¹' (⇑(mk I) '' s) = ⋃ x, (fun y => ↑x + y) '' s
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
|
ext x
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
|
Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s
|
Mathlib_RingTheory_Ideal_Quotient
|
case h
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
I : Ideal R
s : Set R
x : R
⊢ x ∈ ⇑(mk I) ⁻¹' (⇑(mk I) '' s) ↔ x ∈ ⋃ x, (fun y => ↑x + y) '' s
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
|
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
|
Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s
|
Mathlib_RingTheory_Ideal_Quotient
|
case h
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
I : Ideal R
s : Set R
x : R
⊢ (∃ x_1 ∈ s, x_1 - x ∈ I) ↔ ∃ i, ∃ x_1 ∈ s, ↑i + x_1 = x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
|
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
|
Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x✝¹ y : R
I : Ideal R
s : Set R
x : R
x✝ : ∃ x_1 ∈ s, x_1 - x ∈ I
a : R
a_in : a ∈ s
h : a - x ∈ I
⊢ ↑{ val := -(a - x), property := (_ : -(a - x) ∈ I) } + a = x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by
|
simp
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by
|
Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x✝¹ y : R
I : Ideal R
s : Set R
x : R
x✝ : ∃ i, ∃ x_1 ∈ s, ↑i + x_1 = x
i : R
hi : i ∈ I
a : R
ha : a ∈ s
Eq : ↑{ val := i, property := hi } + a = x
⊢ a - x ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by
|
rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by
|
Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x✝¹ y : R
I : Ideal R
s : Set R
x : R
x✝ : ∃ i, ∃ x_1 ∈ s, ↑i + x_1 = x
i : R
hi : i ∈ I
a : R
ha : a ∈ s
Eq : ↑{ val := i, property := hi } + a = x
⊢ -↑{ val := i, property := hi } ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub];
|
exact I.neg_mem hi
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub];
|
Mathlib.RingTheory.Ideal.Quotient.167_0.TwNAv7Pc4PYOWjX
|
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
⊢ IsDomain (R ⧸ I) ↔ IsPrime I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
|
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
|
Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime
|
Mathlib_RingTheory_Ideal_Quotient
|
case refine'_1
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
H : IsDomain (R ⧸ I)
⊢ 0 ≠ 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
·
|
haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
·
|
Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime
|
Mathlib_RingTheory_Ideal_Quotient
|
case refine'_1
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
H : IsDomain (R ⧸ I)
this : Nontrivial (R ⧸ I)
⊢ 0 ≠ 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
|
exact zero_ne_one
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
|
Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime
|
Mathlib_RingTheory_Ideal_Quotient
|
case refine'_2
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
H : IsDomain (R ⧸ I)
x y : R
h : x * y ∈ I
⊢ x ∈ I ∨ y ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
·
|
simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
·
|
Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime
|
Mathlib_RingTheory_Ideal_Quotient
|
case refine'_2
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
H : IsDomain (R ⧸ I)
x y : R
h : (mk I) x * (mk I) y = 0
⊢ (mk I) x = 0 ∨ (mk I) y = 0
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
|
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
|
Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime
|
Mathlib_RingTheory_Ideal_Quotient
|
case refine'_2
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
H : IsDomain (R ⧸ I)
x y : R
h : (mk I) x * (mk I) y = 0
this : NoZeroDivisors (R ⧸ I)
⊢ (mk I) x = 0 ∨ (mk I) y = 0
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
|
exact eq_zero_or_eq_zero_of_mul_eq_zero h
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
|
Mathlib.RingTheory.Ideal.Quotient.189_0.TwNAv7Pc4PYOWjX
|
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
⊢ ∀ {a : R ⧸ I}, a ≠ 0 → ∃ b, a * b = 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
|
rintro ⟨a⟩ h
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
|
Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
⊢ ∃ b, Quot.mk Setoid.r a * b = 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
|
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
|
Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : b * a + c = 1
⊢ ∃ b, Quot.mk Setoid.r a * b = 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
|
rw [mul_comm] at abc
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
|
Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : a * b + c = 1
⊢ ∃ b, Quot.mk Setoid.r a * b = 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
|
refine' ⟨mk _ b, Quot.sound _⟩
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
|
Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : a * b + c = 1
⊢ Setoid.r ((fun x x_1 => x * x_1) a b) 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
|
simp only [Submodule.quotientRel_r_def]
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
|
Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : a * b + c = 1
⊢ a * b - 1 ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
|
rw [← eq_sub_iff_add_eq'] at abc
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
|
Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.intro.intro.intro
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝¹ b✝ : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a✝ : R ⧸ I
a : R
h : Quot.mk Setoid.r a ≠ 0
b c : R
hc : c ∈ I
abc : c = 1 - a * b
⊢ a * b - 1 ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
|
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
|
Mathlib.RingTheory.Ideal.Quotient.198_0.TwNAv7Pc4PYOWjX
|
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a : R ⧸ I
ha : a ≠ 0
⊢ (a * if ha : a = 0 then 0 else choose (_ : ∃ b, a * b = 1)) = 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by
|
rw [dif_neg ha]
|
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by
|
Mathlib.RingTheory.Ideal.Quotient.211_0.TwNAv7Pc4PYOWjX
|
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I)
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a✝ b : R
S : Type v
x y : R
I : Ideal R
hI : IsMaximal I
a : R ⧸ I
ha : a ≠ 0
⊢ a * choose (_ : ∃ b, a * b = 1) = 1
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha];
|
exact Classical.choose_spec (exists_inv ha)
|
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha];
|
Mathlib.RingTheory.Ideal.Quotient.211_0.TwNAv7Pc4PYOWjX
|
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I)
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ IsMaximal I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
|
apply Ideal.isMaximal_iff.2
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ 1 ∉ I ∧ ∀ (J : Ideal R) (x : R), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
|
constructor
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
case left
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ 1 ∉ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
·
|
intro h
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
·
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
case left
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
h : 1 ∈ I
⊢ False
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
|
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
case left.intro.mk.intro.mk
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
hqf : IsField (R ⧸ I)
h : 1 ∈ I
w✝¹ : R ⧸ I
x : R
w✝ : R ⧸ I
y : R
hxy : Quot.mk Setoid.r x ≠ Quot.mk Setoid.r y
⊢ False
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
|
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
case right
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
I : Ideal R
hqf : IsField (R ⧸ I)
⊢ ∀ (J : Ideal R) (x : R), I ≤ J → x ∉ I → x ∈ J → 1 ∈ J
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
·
|
intro J x hIJ hxnI hxJ
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
·
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
case right
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
I : Ideal R
hqf : IsField (R ⧸ I)
J : Ideal R
x : R
hIJ : I ≤ J
hxnI : x ∉ I
hxJ : x ∈ J
⊢ 1 ∈ J
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
|
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
case right.intro.mk
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
hqf : IsField (R ⧸ I)
J : Ideal R
x : R
hIJ : I ≤ J
hxnI : x ∉ I
hxJ : x ∈ J
w✝ : R ⧸ I
y : R
hy : (mk I) x * Quot.mk Setoid.r y = 1
⊢ 1 ∈ J
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
|
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
case right.intro.mk
R : Type u
inst✝ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y✝ : R
I : Ideal R
hqf : IsField (R ⧸ I)
J : Ideal R
x : R
hIJ : I ≤ J
hxnI : x ∉ I
hxJ : x ∈ J
w✝ : R ⧸ I
y : R
hy : (mk I) x * Quot.mk Setoid.r y = 1
⊢ x * y - (x * y - 1) ∈ J
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
|
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
|
Mathlib.RingTheory.Ideal.Quotient.233_0.TwNAv7Pc4PYOWjX
|
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝¹ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y : R
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
hf : Function.Surjective ⇑f
⊢ Function.Surjective ⇑(lift I f H)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
|
intro y
|
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
|
Mathlib.RingTheory.Ideal.Quotient.270_0.TwNAv7Pc4PYOWjX
|
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H)
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝¹ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x y✝ : R
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
hf : Function.Surjective ⇑f
y : S
⊢ ∃ a, (lift I f H) a = y
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
|
obtain ⟨x, rfl⟩ := hf y
|
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
|
Mathlib.RingTheory.Ideal.Quotient.270_0.TwNAv7Pc4PYOWjX
|
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H)
|
Mathlib_RingTheory_Ideal_Quotient
|
case intro
R : Type u
inst✝¹ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
hf : Function.Surjective ⇑f
x : R
⊢ ∃ a, (lift I f H) a = f x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
|
use Ideal.Quotient.mk I x
|
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
|
Mathlib.RingTheory.Ideal.Quotient.270_0.TwNAv7Pc4PYOWjX
|
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H)
|
Mathlib_RingTheory_Ideal_Quotient
|
case h
R : Type u
inst✝¹ : CommRing R
I✝ : Ideal R
a b : R
S : Type v
x✝ y : R
inst✝ : CommRing S
I : Ideal R
f : R →+* S
H : ∀ a ∈ I, f a = 0
hf : Function.Surjective ⇑f
x : R
⊢ (lift I f H) ((mk I) x) = f x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
|
simp only [Ideal.Quotient.lift_mk]
|
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
|
Mathlib.RingTheory.Ideal.Quotient.270_0.TwNAv7Pc4PYOWjX
|
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H)
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S✝ : Type v
x y : R
inst✝ : CommRing S✝
S T : Ideal R
H : S ≤ T
⊢ RingHom.comp (factor S T H) (mk S) = mk T
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
|
ext x
|
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
|
Mathlib.RingTheory.Ideal.Quotient.290_0.TwNAv7Pc4PYOWjX
|
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T
|
Mathlib_RingTheory_Ideal_Quotient
|
case a
R : Type u
inst✝¹ : CommRing R
I : Ideal R
a b : R
S✝ : Type v
x✝ y : R
inst✝ : CommRing S✝
S T : Ideal R
H : S ≤ T
x : R
⊢ (RingHom.comp (factor S T H) (mk S)) x = (mk T) x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
|
rw [RingHom.comp_apply, factor_mk]
|
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
|
Mathlib.RingTheory.Ideal.Quotient.290_0.TwNAv7Pc4PYOWjX
|
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T
|
Mathlib_RingTheory_Ideal_Quotient
|
R✝ : Type u
inst✝¹ : CommRing R✝
I✝ : Ideal R✝
a b : R✝
S : Type v
R : Type u_1
inst✝ : CommRing R
I J : Ideal R
h : I = J
src✝ : (R ⧸ I) ≃ₗ[R] R ⧸ J := Submodule.quotEquivOfEq I J h
⊢ ∀ (x y : R ⧸ I),
Equiv.toFun
{ toFun := src✝.toFun, invFun := src✝.invFun, left_inv := (_ : Function.LeftInverse src✝.invFun src✝.toFun),
right_inv := (_ : Function.RightInverse src✝.invFun src✝.toFun) }
(x * y) =
Equiv.toFun
{ toFun := src✝.toFun, invFun := src✝.invFun, left_inv := (_ : Function.LeftInverse src✝.invFun src✝.toFun),
right_inv := (_ : Function.RightInverse src✝.invFun src✝.toFun) }
x *
Equiv.toFun
{ toFun := src✝.toFun, invFun := src✝.invFun, left_inv := (_ : Function.LeftInverse src✝.invFun src✝.toFun),
right_inv := (_ : Function.RightInverse src✝.invFun src✝.toFun) }
y
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
|
rintro ⟨x⟩ ⟨y⟩
|
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
|
Mathlib.RingTheory.Ideal.Quotient.298_0.TwNAv7Pc4PYOWjX
|
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk
R✝ : Type u
inst✝¹ : CommRing R✝
I✝ : Ideal R✝
a b : R✝
S : Type v
R : Type u_1
inst✝ : CommRing R
I J : Ideal R
h : I = J
src✝ : (R ⧸ I) ≃ₗ[R] R ⧸ J := Submodule.quotEquivOfEq I J h
x✝ : R ⧸ I
x : R
y✝ : R ⧸ I
y : R
⊢ Equiv.toFun
{ toFun := src✝.toFun, invFun := src✝.invFun, left_inv := (_ : Function.LeftInverse src✝.invFun src✝.toFun),
right_inv := (_ : Function.RightInverse src✝.invFun src✝.toFun) }
(Quot.mk Setoid.r x * Quot.mk Setoid.r y) =
Equiv.toFun
{ toFun := src✝.toFun, invFun := src✝.invFun, left_inv := (_ : Function.LeftInverse src✝.invFun src✝.toFun),
right_inv := (_ : Function.RightInverse src✝.invFun src✝.toFun) }
(Quot.mk Setoid.r x) *
Equiv.toFun
{ toFun := src✝.toFun, invFun := src✝.invFun, left_inv := (_ : Function.LeftInverse src✝.invFun src✝.toFun),
right_inv := (_ : Function.RightInverse src✝.invFun src✝.toFun) }
(Quot.mk Setoid.r y)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
|
rfl
|
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
|
Mathlib.RingTheory.Ideal.Quotient.298_0.TwNAv7Pc4PYOWjX
|
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J
|
Mathlib_RingTheory_Ideal_Quotient
|
R✝ : Type u
inst✝¹ : CommRing R✝
I✝ : Ideal R✝
a b : R✝
S : Type v
R : Type u_1
inst✝ : CommRing R
I J : Ideal R
h : I = J
⊢ RingEquiv.symm (quotEquivOfEq h) = quotEquivOfEq (_ : J = I)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by
|
ext
|
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by
|
Mathlib.RingTheory.Ideal.Quotient.315_0.TwNAv7Pc4PYOWjX
|
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm
|
Mathlib_RingTheory_Ideal_Quotient
|
case h
R✝ : Type u
inst✝¹ : CommRing R✝
I✝ : Ideal R✝
a b : R✝
S : Type v
R : Type u_1
inst✝ : CommRing R
I J : Ideal R
h : I = J
x✝ : R ⧸ J
⊢ (RingEquiv.symm (quotEquivOfEq h)) x✝ = (quotEquivOfEq (_ : J = I)) x✝
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext;
|
rfl
|
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext;
|
Mathlib.RingTheory.Ideal.Quotient.315_0.TwNAv7Pc4PYOWjX
|
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
⊢ ∀ (a₁ : R) (a₂ : ι → R) (b₁ : R) (b₂ : ι → R),
Setoid.r a₁ b₁ →
Setoid.r a₂ b₂ →
(fun r m => Submodule.Quotient.mk (r • m)) a₁ a₂ = (fun r m => Submodule.Quotient.mk (r • m)) b₁ b₂
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
|
intro c₁ m₁ c₂ m₂ hc hm
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
c₁ : R
m₁ : ι → R
c₂ : R
m₂ : ι → R
hc : Setoid.r c₁ c₂
hm : Setoid.r m₁ m₂
⊢ (fun r m => Submodule.Quotient.mk (r • m)) c₁ m₁ = (fun r m => Submodule.Quotient.mk (r • m)) c₂ m₂
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
|
apply Ideal.Quotient.eq.2
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
c₁ : R
m₁ : ι → R
c₂ : R
m₂ : ι → R
hc : Setoid.r c₁ c₂
hm : Setoid.r m₁ m₂
⊢ c₁ • m₁ - c₂ • m₂ ∈ pi I ι
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
|
rw [Submodule.quotientRel_r_def] at hc hm
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
c₁ : R
m₁ : ι → R
c₂ : R
m₂ : ι → R
hc : c₁ - c₂ ∈ I
hm : m₁ - m₂ ∈ pi I ι
⊢ c₁ • m₁ - c₂ • m₂ ∈ pi I ι
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
|
intro i
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
c : R ⧸ I
m : (ι → R) ⧸ pi I ι
c₁ : R
m₁ : ι → R
c₂ : R
m₂ : ι → R
hc : c₁ - c₂ ∈ I
hm : m₁ - m₂ ∈ pi I ι
i : ι
⊢ (c₁ • m₁ - c₂ • m₂) i ∈ I
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
|
exact I.mul_sub_mul_mem hc (hm i)
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (b : (ι → R) ⧸ pi I ι), 1 • b = b
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
|
rintro ⟨a⟩
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
b✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ 1 • Quot.mk Setoid.r a = Quot.mk Setoid.r a
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
|
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
b✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ (Quotient.mk (pi I ι)) (1 • a) = (Quotient.mk (pi I ι)) a
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
congr with i
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
b✝ : (ι → R) ⧸ pi I ι
a : ι → R
i : ι
⊢ (1 • a) i = a i
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i;
|
exact one_mul (a i)
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i;
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (x y : R ⧸ I) (b : (ι → R) ⧸ pi I ι), (x * y) • b = x • y • b
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
|
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝¹ : R
S ι : Type v
x✝ : R ⧸ I
a : R
y✝ : R ⧸ I
b : R
b✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quot.mk Setoid.r a * Quot.mk Setoid.r b) • Quot.mk Setoid.r c =
Quot.mk Setoid.r a • Quot.mk Setoid.r b • Quot.mk Setoid.r c
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝¹ : R
S ι : Type v
x✝ : R ⧸ I
a : R
y✝ : R ⧸ I
b : R
b✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quotient.mk (pi I ι)) ((fun x x_1 => x * x_1) a b • c) = (Quotient.mk (pi I ι)) (a • b • c)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
congr 1
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk.convert_3.h.e_6.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝¹ : R
S ι : Type v
x✝ : R ⧸ I
a : R
y✝ : R ⧸ I
b : R
b✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (fun x x_1 => x * x_1) a b • c = a • b • c
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1;
|
funext i
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1;
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝¹ : R
S ι : Type v
x✝ : R ⧸ I
a : R
y✝ : R ⧸ I
b : R
b✝ : (ι → R) ⧸ pi I ι
c : ι → R
i : ι
⊢ ((fun x x_1 => x * x_1) a b • c) i = (a • b • c) i
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i;
|
exact mul_assoc a b (c i)
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i;
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (a : R ⧸ I), a • 0 = 0
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
|
rintro ⟨a⟩
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b : R
S ι : Type v
a✝ : R ⧸ I
a : R
⊢ Quot.mk Setoid.r a • 0 = 0
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
|
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b : R
S ι : Type v
a✝ : R ⧸ I
a : R
⊢ (Quotient.mk (pi I ι)) (a • 0) = (Quotient.mk (pi I ι)) 0
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
congr with _
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b : R
S ι : Type v
a✝ : R ⧸ I
a : R
x✝ : ι
⊢ (a • 0) x✝ = OfNat.ofNat 0 x✝
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _;
|
exact mul_zero a
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _;
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (a : R ⧸ I) (x y : (ι → R) ⧸ pi I ι), a • (x + y) = a • x + a • y
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
|
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b✝ : R
S ι : Type v
a✝ : R ⧸ I
a : R
x✝ : (ι → R) ⧸ pi I ι
b : ι → R
y✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ Quot.mk Setoid.r a • (Quot.mk Setoid.r b + Quot.mk Setoid.r c) =
Quot.mk Setoid.r a • Quot.mk Setoid.r b + Quot.mk Setoid.r a • Quot.mk Setoid.r c
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b✝ : R
S ι : Type v
a✝ : R ⧸ I
a : R
x✝ : (ι → R) ⧸ pi I ι
b : ι → R
y✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quotient.mk (pi I ι)) (a • (fun x x_1 => x + x_1) b c) =
(Quotient.mk (pi I ι)) ((fun x x_1 => x + x_1) (a • b) (a • c))
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
congr with i
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝¹ b✝ : R
S ι : Type v
a✝ : R ⧸ I
a : R
x✝ : (ι → R) ⧸ pi I ι
b : ι → R
y✝ : (ι → R) ⧸ pi I ι
c : ι → R
i : ι
⊢ (a • (fun x x_1 => x + x_1) b c) i = (fun x x_1 => x + x_1) (a • b) (a • c) i
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i;
|
exact mul_add a (b i) (c i)
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i;
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (r s : R ⧸ I) (x : (ι → R) ⧸ pi I ι), (r + s) • x = r • x + s • x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
|
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝ : R
S ι : Type v
r✝ : R ⧸ I
a : R
s✝ : R ⧸ I
b : R
x✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quot.mk Setoid.r a + Quot.mk Setoid.r b) • Quot.mk Setoid.r c =
Quot.mk Setoid.r a • Quot.mk Setoid.r c + Quot.mk Setoid.r b • Quot.mk Setoid.r c
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝ : R
S ι : Type v
r✝ : R ⧸ I
a : R
s✝ : R ⧸ I
b : R
x✝ : (ι → R) ⧸ pi I ι
c : ι → R
⊢ (Quotient.mk (pi I ι)) ((fun x x_1 => x + x_1) a b • c) =
(Quotient.mk (pi I ι)) ((fun x x_1 => x + x_1) (a • c) (b • c))
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
congr with i
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk.mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b✝ : R
S ι : Type v
r✝ : R ⧸ I
a : R
s✝ : R ⧸ I
b : R
x✝ : (ι → R) ⧸ pi I ι
c : ι → R
i : ι
⊢ ((fun x x_1 => x + x_1) a b • c) i = (fun x x_1 => x + x_1) (a • c) (b • c) i
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i;
|
exact add_mul a b (c i)
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i;
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (x : (ι → R) ⧸ pi I ι), 0 • x = 0
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
|
rintro ⟨a⟩
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ 0 • Quot.mk Setoid.r a = 0
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
|
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.convert_3
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
⊢ (Quotient.mk (pi I ι)) (0 • a) = (Quotient.mk (pi I ι)) 0
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
congr with i
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.convert_3.h.e_6.h.h
R : Type u
inst✝ : CommRing R
I : Ideal R
a✝ b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a : ι → R
i : ι
⊢ (0 • a) i = OfNat.ofNat 0 i
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i;
|
exact zero_mul (a i)
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i;
|
Mathlib.RingTheory.Ideal.Quotient.324_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact zero_mul (a i)
#align ideal.module_pi Ideal.modulePi
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by
|
rintro ⟨_⟩ ⟨_⟩
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by
|
Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
a✝¹ : ι → R
y✝ : (ι → R) ⧸ pi I ι
a✝ : ι → R
⊢ (fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(Quot.mk Setoid.r a✝¹ + Quot.mk Setoid.r a✝) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(Quot.mk Setoid.r a✝¹) +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(Quot.mk Setoid.r a✝)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact zero_mul (a i)
#align ideal.module_pi Ideal.modulePi
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩;
|
rfl
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩;
|
Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ ∀ (r : R ⧸ I) (x : (ι → R) ⧸ pi I ι),
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y) }
(r • x) =
(RingHom.id (R ⧸ I)) r •
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y) }
x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact zero_mul (a i)
#align ideal.module_pi Ideal.modulePi
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by
|
rintro ⟨_⟩ ⟨_⟩
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by
|
Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk.mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
r✝ : R ⧸ I
a✝¹ : R
x✝ : (ι → R) ⧸ pi I ι
a✝ : ι → R
⊢ AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y) }
(Quot.mk Setoid.r a✝¹ • Quot.mk Setoid.r a✝) =
(RingHom.id (R ⧸ I)) (Quot.mk Setoid.r a✝¹) •
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y) }
(Quot.mk Setoid.r a✝)
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact zero_mul (a i)
#align ideal.module_pi Ideal.modulePi
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩;
|
rfl
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩;
|
Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ Function.LeftInverse (fun x => (Quotient.mk (pi I ι)) fun i => Quotient.out' (x i))
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y) },
map_smul' :=
(_ :
∀ (r : R ⧸ I) (x : (ι → R) ⧸ pi I ι),
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
y) }
(r • x) =
(RingHom.id (R ⧸ I)) r •
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
y) }
x) }.toAddHom.toFun
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact zero_mul (a i)
#align ideal.module_pi Ideal.modulePi
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩; rfl
invFun := fun x ↦ Ideal.Quotient.mk (I.pi ι) fun i ↦ Quotient.out' (x i)
left_inv := by
|
rintro ⟨x⟩
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩; rfl
invFun := fun x ↦ Ideal.Quotient.mk (I.pi ι) fun i ↦ Quotient.out' (x i)
left_inv := by
|
Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun
|
Mathlib_RingTheory_Ideal_Quotient
|
case mk
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x✝ : (ι → R) ⧸ pi I ι
x : ι → R
⊢ (fun x => (Quotient.mk (pi I ι)) fun i => Quotient.out' (x i))
(AddHom.toFun
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y) },
map_smul' :=
(_ :
∀ (r : R ⧸ I) (x : (ι → R) ⧸ pi I ι),
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
y) }
(r • x) =
(RingHom.id (R ⧸ I)) r •
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
y) }
x) }.toAddHom
(Quot.mk Setoid.r x)) =
Quot.mk Setoid.r x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact zero_mul (a i)
#align ideal.module_pi Ideal.modulePi
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩; rfl
invFun := fun x ↦ Ideal.Quotient.mk (I.pi ι) fun i ↦ Quotient.out' (x i)
left_inv := by
rintro ⟨x⟩
|
exact Ideal.Quotient.eq.2 fun i => Ideal.Quotient.eq.1 (Quotient.out_eq' _)
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩; rfl
invFun := fun x ↦ Ideal.Quotient.mk (I.pi ι) fun i ↦ Quotient.out' (x i)
left_inv := by
rintro ⟨x⟩
|
Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
⊢ Function.RightInverse (fun x => (Quotient.mk (pi I ι)) fun i => Quotient.out' (x i))
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y) },
map_smul' :=
(_ :
∀ (r : R ⧸ I) (x : (ι → R) ⧸ pi I ι),
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
y) }
(r • x) =
(RingHom.id (R ⧸ I)) r •
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
y) }
x) }.toAddHom.toFun
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact zero_mul (a i)
#align ideal.module_pi Ideal.modulePi
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩; rfl
invFun := fun x ↦ Ideal.Quotient.mk (I.pi ι) fun i ↦ Quotient.out' (x i)
left_inv := by
rintro ⟨x⟩
exact Ideal.Quotient.eq.2 fun i => Ideal.Quotient.eq.1 (Quotient.out_eq' _)
right_inv := by
|
intro x
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩; rfl
invFun := fun x ↦ Ideal.Quotient.mk (I.pi ι) fun i ↦ Quotient.out' (x i)
left_inv := by
rintro ⟨x⟩
exact Ideal.Quotient.eq.2 fun i => Ideal.Quotient.eq.1 (Quotient.out_eq' _)
right_inv := by
|
Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun
|
Mathlib_RingTheory_Ideal_Quotient
|
R : Type u
inst✝ : CommRing R
I : Ideal R
a b : R
S ι : Type v
x : ι → R ⧸ I
⊢ AddHom.toFun
{
toAddHom :=
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b → (fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
y) },
map_smul' :=
(_ :
∀ (r : R ⧸ I) (x : (ι → R) ⧸ pi I ι),
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
y) }
(r • x) =
(RingHom.id (R ⧸ I)) r •
AddHom.toFun
{
toFun := fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a = (fun f i => (Quotient.mk I) (f i)) b),
map_add' :=
(_ :
∀ (x y : (ι → R) ⧸ pi I ι),
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
(x + y) =
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
x +
(fun x =>
Quotient.liftOn' x (fun f i => (Quotient.mk I) (f i))
(_ :
∀ (a b : ι → R),
Setoid.r a b →
(fun f i => (Quotient.mk I) (f i)) a =
(fun f i => (Quotient.mk I) (f i)) b))
y) }
x) }.toAddHom
((fun x => (Quotient.mk (pi I ι)) fun i => Quotient.out' (x i)) x) =
x
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.Ring.Fin
import Mathlib.Algebra.Ring.Prod
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
/-!
# Ideal quotients
This file defines ideal quotients as a special case of submodule quotients and proves some basic
results about these quotients.
See `Algebra.RingQuot` for quotients of non-commutative rings.
## Main definitions
- `Ideal.Quotient`: the quotient of a commutative ring `R` by an ideal `I : Ideal R`
-/
universe u v w
namespace Ideal
open Set
open BigOperators
variable {R : Type u} [CommRing R] (I : Ideal R) {a b : R}
variable {S : Type v}
-- Note that at present `Ideal` means a left-ideal,
-- so this quotient is only useful in a commutative ring.
-- We should develop quotients by two-sided ideals as well.
/-- The quotient `R/I` of a ring `R` by an ideal `I`.
The ideal quotient of `I` is defined to equal the quotient of `I` as an `R`-submodule of `R`.
This definition is marked `reducible` so that typeclass instances can be shared between
`Ideal.Quotient I` and `Submodule.Quotient I`.
-/
@[reducible]
instance : HasQuotient R (Ideal R) :=
Submodule.hasQuotient
namespace Quotient
variable {I} {x y : R}
instance one (I : Ideal R) : One (R ⧸ I) :=
⟨Submodule.Quotient.mk 1⟩
#align ideal.quotient.has_one Ideal.Quotient.one
/-- On `Ideal`s, `Submodule.quotientRel` is a ring congruence. -/
protected def ringCon (I : Ideal R) : RingCon R :=
{ QuotientAddGroup.con I.toAddSubgroup with
mul' := fun {a₁ b₁ a₂ b₂} h₁ h₂ => by
rw [Submodule.quotientRel_r_def] at h₁ h₂ ⊢
have F := I.add_mem (I.mul_mem_left a₂ h₁) (I.mul_mem_right b₁ h₂)
have : a₁ * a₂ - b₁ * b₂ = a₂ * (a₁ - b₁) + (a₂ - b₂) * b₁ := by
rw [mul_sub, sub_mul, sub_add_sub_cancel, mul_comm, mul_comm b₁]
rwa [← this] at F }
#align ideal.quotient.ring_con Ideal.Quotient.ringCon
instance commRing (I : Ideal R) : CommRing (R ⧸ I) :=
inferInstanceAs (CommRing (Quotient.ringCon I).Quotient)
#align ideal.quotient.comm_ring Ideal.Quotient.commRing
-- Sanity test to make sure no diamonds have emerged in `commRing`
example : (commRing I).toAddCommGroup = Submodule.Quotient.addCommGroup I := rfl
-- this instance is harder to find than the one via `Algebra α (R ⧸ I)`, so use a lower priority
instance (priority := 100) isScalarTower_right {α} [SMul α R] [IsScalarTower α R R] :
IsScalarTower α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).isScalarTower_right
#align ideal.quotient.is_scalar_tower_right Ideal.Quotient.isScalarTower_right
instance smulCommClass {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass α R R] :
SMulCommClass α (R ⧸ I) (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass
#align ideal.quotient.smul_comm_class Ideal.Quotient.smulCommClass
instance smulCommClass' {α} [SMul α R] [IsScalarTower α R R] [SMulCommClass R α R] :
SMulCommClass (R ⧸ I) α (R ⧸ I) :=
(Quotient.ringCon I).smulCommClass'
#align ideal.quotient.smul_comm_class' Ideal.Quotient.smulCommClass'
/-- The ring homomorphism from a ring `R` to a quotient ring `R/I`. -/
def mk (I : Ideal R) : R →+* R ⧸ I where
toFun a := Submodule.Quotient.mk a
map_zero' := rfl
map_one' := rfl
map_mul' _ _ := rfl
map_add' _ _ := rfl
#align ideal.quotient.mk Ideal.Quotient.mk
instance {I : Ideal R} : Coe R (R ⧸ I) :=
⟨Ideal.Quotient.mk I⟩
/-- Two `RingHom`s from the quotient by an ideal are equal if their
compositions with `Ideal.Quotient.mk'` are equal.
See note [partially-applied ext lemmas]. -/
@[ext 1100]
theorem ringHom_ext [NonAssocSemiring S] ⦃f g : R ⧸ I →+* S⦄ (h : f.comp (mk I) = g.comp (mk I)) :
f = g :=
RingHom.ext fun x => Quotient.inductionOn' x <| (RingHom.congr_fun h : _)
#align ideal.quotient.ring_hom_ext Ideal.Quotient.ringHom_ext
instance inhabited : Inhabited (R ⧸ I) :=
⟨mk I 37⟩
#align ideal.quotient.inhabited Ideal.Quotient.inhabited
protected theorem eq : mk I x = mk I y ↔ x - y ∈ I :=
Submodule.Quotient.eq I
#align ideal.quotient.eq Ideal.Quotient.eq
@[simp]
theorem mk_eq_mk (x : R) : (Submodule.Quotient.mk x : R ⧸ I) = mk I x := rfl
#align ideal.quotient.mk_eq_mk Ideal.Quotient.mk_eq_mk
theorem eq_zero_iff_mem {I : Ideal R} : mk I a = 0 ↔ a ∈ I :=
Submodule.Quotient.mk_eq_zero _
#align ideal.quotient.eq_zero_iff_mem Ideal.Quotient.eq_zero_iff_mem
theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
-- Porting note: new theorem
theorem mk_eq_mk_iff_sub_mem (x y : R) : mk I x = mk I y ↔ x - y ∈ I := by
rw [← eq_zero_iff_mem, map_sub, sub_eq_zero]
theorem zero_eq_one_iff {I : Ideal R} : (0 : R ⧸ I) = 1 ↔ I = ⊤ :=
eq_comm.trans <| eq_zero_iff_mem.trans (eq_top_iff_one _).symm
#align ideal.quotient.zero_eq_one_iff Ideal.Quotient.zero_eq_one_iff
theorem zero_ne_one_iff {I : Ideal R} : (0 : R ⧸ I) ≠ 1 ↔ I ≠ ⊤ :=
not_congr zero_eq_one_iff
#align ideal.quotient.zero_ne_one_iff Ideal.Quotient.zero_ne_one_iff
protected theorem nontrivial {I : Ideal R} (hI : I ≠ ⊤) : Nontrivial (R ⧸ I) :=
⟨⟨0, 1, zero_ne_one_iff.2 hI⟩⟩
#align ideal.quotient.nontrivial Ideal.Quotient.nontrivial
theorem subsingleton_iff {I : Ideal R} : Subsingleton (R ⧸ I) ↔ I = ⊤ := by
rw [eq_top_iff_one, ← subsingleton_iff_zero_eq_one, eq_comm, ← (mk I).map_one,
Quotient.eq_zero_iff_mem]
#align ideal.quotient.subsingleton_iff Ideal.Quotient.subsingleton_iff
instance : Unique (R ⧸ (⊤ : Ideal R)) :=
⟨⟨0⟩, by rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top⟩
theorem mk_surjective : Function.Surjective (mk I) := fun y =>
Quotient.inductionOn' y fun x => Exists.intro x rfl
#align ideal.quotient.mk_surjective Ideal.Quotient.mk_surjective
instance : RingHomSurjective (mk I) :=
⟨mk_surjective⟩
/-- If `I` is an ideal of a commutative ring `R`, if `q : R → R/I` is the quotient map, and if
`s ⊆ R` is a subset, then `q⁻¹(q(s)) = ⋃ᵢ(i + s)`, the union running over all `i ∈ I`. -/
theorem quotient_ring_saturate (I : Ideal R) (s : Set R) :
mk I ⁻¹' (mk I '' s) = ⋃ x : I, (fun y => x.1 + y) '' s := by
ext x
simp only [mem_preimage, mem_image, mem_iUnion, Ideal.Quotient.eq]
exact
⟨fun ⟨a, a_in, h⟩ => ⟨⟨_, I.neg_mem h⟩, a, a_in, by simp⟩, fun ⟨⟨i, hi⟩, a, ha, Eq⟩ =>
⟨a, ha, by rw [← Eq, sub_add_eq_sub_sub_swap, sub_self, zero_sub]; exact I.neg_mem hi⟩⟩
#align ideal.quotient.quotient_ring_saturate Ideal.Quotient.quotient_ring_saturate
instance noZeroDivisors (I : Ideal R) [hI : I.IsPrime] : NoZeroDivisors (R ⧸ I) where
eq_zero_or_eq_zero_of_mul_eq_zero {a b} := Quotient.inductionOn₂' a b fun {_ _} hab =>
(hI.mem_or_mem (eq_zero_iff_mem.1 hab)).elim (Or.inl ∘ eq_zero_iff_mem.2)
(Or.inr ∘ eq_zero_iff_mem.2)
#align ideal.quotient.no_zero_divisors Ideal.Quotient.noZeroDivisors
instance isDomain (I : Ideal R) [hI : I.IsPrime] : IsDomain (R ⧸ I) :=
let _ := Quotient.nontrivial hI.1
NoZeroDivisors.to_isDomain _
#align ideal.quotient.is_domain Ideal.Quotient.isDomain
theorem isDomain_iff_prime (I : Ideal R) : IsDomain (R ⧸ I) ↔ I.IsPrime := by
refine' ⟨fun H => ⟨zero_ne_one_iff.1 _, fun {x y} h => _⟩, fun h => inferInstance⟩
· haveI : Nontrivial (R ⧸ I) := ⟨H.2.1⟩
exact zero_ne_one
· simp only [← eq_zero_iff_mem, (mk I).map_mul] at h ⊢
haveI := @IsDomain.to_noZeroDivisors (R ⧸ I) _ H
exact eq_zero_or_eq_zero_of_mul_eq_zero h
#align ideal.quotient.is_domain_iff_prime Ideal.Quotient.isDomain_iff_prime
theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
∀ {a : R ⧸ I}, a ≠ 0 → ∃ b : R ⧸ I, a * b = 1 := by
rintro ⟨a⟩ h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with ⟨b, c, hc, abc⟩
rw [mul_comm] at abc
refine' ⟨mk _ b, Quot.sound _⟩
simp only [Submodule.quotientRel_r_def]
rw [← eq_sub_iff_add_eq'] at abc
rwa [abc, ← neg_mem_iff (G := R) (H := I), neg_sub] at hc
#align ideal.quotient.exists_inv Ideal.Quotient.exists_inv
open Classical
/-- The quotient by a maximal ideal is a group with zero. This is a `def` rather than `instance`,
since users will have computable inverses in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def groupWithZero (I : Ideal R) [hI : I.IsMaximal] :
GroupWithZero (R ⧸ I) :=
{ inv := fun a => if ha : a = 0 then 0 else Classical.choose (exists_inv ha)
mul_inv_cancel := fun a (ha : a ≠ 0) =>
show a * dite _ _ _ = _ by rw [dif_neg ha]; exact Classical.choose_spec (exists_inv ha)
inv_zero := dif_pos rfl }
#align ideal.quotient.group_with_zero Ideal.Quotient.groupWithZero
/-- The quotient by a maximal ideal is a field. This is a `def` rather than `instance`, since users
will have computable inverses (and `qsmul`, `rat_cast`) in some applications.
See note [reducible non-instances]. -/
@[reducible]
protected noncomputable def field (I : Ideal R) [hI : I.IsMaximal] : Field (R ⧸ I) :=
{ Quotient.commRing I, Quotient.groupWithZero I with }
#align ideal.quotient.field Ideal.Quotient.field
/-- If the quotient by an ideal is a field, then the ideal is maximal. -/
theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 : R), ← sub_self (x * y), sub_add]
refine' J.sub_mem (J.mul_mem_right _ hxJ) (hIJ (Ideal.Quotient.eq.1 hy))
#align ideal.quotient.maximal_of_is_field Ideal.Quotient.maximal_of_isField
/-- The quotient of a ring by an ideal is a field iff the ideal is maximal. -/
theorem maximal_ideal_iff_isField_quotient (I : Ideal R) : I.IsMaximal ↔ IsField (R ⧸ I) :=
⟨fun h =>
let _i := @Quotient.field _ _ I h
Field.toIsField _,
maximal_of_isField _⟩
#align ideal.quotient.maximal_ideal_iff_is_field_quotient Ideal.Quotient.maximal_ideal_iff_isField_quotient
variable [CommRing S]
/-- Given a ring homomorphism `f : R →+* S` sending all elements of an ideal to zero,
lift it to the quotient by this ideal. -/
def lift (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) : R ⧸ I →+* S :=
{ QuotientAddGroup.lift I.toAddSubgroup f.toAddMonoidHom H with
map_one' := f.map_one
map_mul' := fun a₁ a₂ => Quotient.inductionOn₂' a₁ a₂ f.map_mul }
#align ideal.quotient.lift Ideal.Quotient.lift
@[simp]
theorem lift_mk (I : Ideal R) (f : R →+* S) (H : ∀ a : R, a ∈ I → f a = 0) :
lift I f H (mk I a) = f a :=
rfl
#align ideal.quotient.lift_mk Ideal.Quotient.lift_mk
theorem lift_surjective_of_surjective (I : Ideal R) {f : R →+* S} (H : ∀ a : R, a ∈ I → f a = 0)
(hf : Function.Surjective f) : Function.Surjective (Ideal.Quotient.lift I f H) := by
intro y
obtain ⟨x, rfl⟩ := hf y
use Ideal.Quotient.mk I x
simp only [Ideal.Quotient.lift_mk]
#align ideal.quotient.lift_surjective_of_surjective Ideal.Quotient.lift_surjective_of_surjective
/-- The ring homomorphism from the quotient by a smaller ideal to the quotient by a larger ideal.
This is the `Ideal.Quotient` version of `Quot.Factor` -/
def factor (S T : Ideal R) (H : S ≤ T) : R ⧸ S →+* R ⧸ T :=
Ideal.Quotient.lift S (mk T) fun _ hx => eq_zero_iff_mem.2 (H hx)
#align ideal.quotient.factor Ideal.Quotient.factor
@[simp]
theorem factor_mk (S T : Ideal R) (H : S ≤ T) (x : R) : factor S T H (mk S x) = mk T x :=
rfl
#align ideal.quotient.factor_mk Ideal.Quotient.factor_mk
@[simp]
theorem factor_comp_mk (S T : Ideal R) (H : S ≤ T) : (factor S T H).comp (mk S) = mk T := by
ext x
rw [RingHom.comp_apply, factor_mk]
#align ideal.quotient.factor_comp_mk Ideal.Quotient.factor_comp_mk
end Quotient
/-- Quotienting by equal ideals gives equivalent rings.
See also `Submodule.quotEquivOfEq` and `Ideal.quotientEquivAlgOfEq`.
-/
def quotEquivOfEq {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) : R ⧸ I ≃+* R ⧸ J :=
{ Submodule.quotEquivOfEq I J h with
map_mul' := by
rintro ⟨x⟩ ⟨y⟩
rfl }
#align ideal.quot_equiv_of_eq Ideal.quotEquivOfEq
@[simp]
theorem quotEquivOfEq_mk {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) (x : R) :
quotEquivOfEq h (Ideal.Quotient.mk I x) = Ideal.Quotient.mk J x :=
rfl
#align ideal.quot_equiv_of_eq_mk Ideal.quotEquivOfEq_mk
@[simp]
theorem quotEquivOfEq_symm {R : Type*} [CommRing R] {I J : Ideal R} (h : I = J) :
(Ideal.quotEquivOfEq h).symm = Ideal.quotEquivOfEq h.symm := by ext; rfl
#align ideal.quot_equiv_of_eq_symm Ideal.quotEquivOfEq_symm
section Pi
variable (ι : Type v)
/-- `R^n/I^n` is a `R/I`-module. -/
instance modulePi : Module (R ⧸ I) ((ι → R) ⧸ I.pi ι) where
smul c m :=
Quotient.liftOn₂' c m (fun r m => Submodule.Quotient.mk <| r • m) $ by
intro c₁ m₁ c₂ m₂ hc hm
apply Ideal.Quotient.eq.2
rw [Submodule.quotientRel_r_def] at hc hm
intro i
exact I.mul_sub_mul_mem hc (hm i)
one_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact one_mul (a i)
mul_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr 1; funext i; exact mul_assoc a b (c i)
smul_add := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact mul_add a (b i) (c i)
smul_zero := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with _; exact mul_zero a
add_smul := by
rintro ⟨a⟩ ⟨b⟩ ⟨c⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact add_mul a b (c i)
zero_smul := by
rintro ⟨a⟩
convert_to Ideal.Quotient.mk (I.pi ι) _ = Ideal.Quotient.mk (I.pi ι) _
congr with i; exact zero_mul (a i)
#align ideal.module_pi Ideal.modulePi
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩; rfl
invFun := fun x ↦ Ideal.Quotient.mk (I.pi ι) fun i ↦ Quotient.out' (x i)
left_inv := by
rintro ⟨x⟩
exact Ideal.Quotient.eq.2 fun i => Ideal.Quotient.eq.1 (Quotient.out_eq' _)
right_inv := by
intro x
|
ext i
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun := fun x ↦
Quotient.liftOn' x (fun f i => Ideal.Quotient.mk I (f i)) fun a b hab =>
funext fun i => (Submodule.Quotient.eq' _).2 (QuotientAddGroup.leftRel_apply.mp hab i)
map_add' := by rintro ⟨_⟩ ⟨_⟩; rfl
map_smul' := by rintro ⟨_⟩ ⟨_⟩; rfl
invFun := fun x ↦ Ideal.Quotient.mk (I.pi ι) fun i ↦ Quotient.out' (x i)
left_inv := by
rintro ⟨x⟩
exact Ideal.Quotient.eq.2 fun i => Ideal.Quotient.eq.1 (Quotient.out_eq' _)
right_inv := by
intro x
|
Mathlib.RingTheory.Ideal.Quotient.359_0.TwNAv7Pc4PYOWjX
|
/-- `R^n/I^n` is isomorphic to `(R/I)^n` as an `R/I`-module. -/
noncomputable def piQuotEquiv : ((ι → R) ⧸ I.pi ι) ≃ₗ[R ⧸ I] ι → (R ⧸ I) where
toFun
|
Mathlib_RingTheory_Ideal_Quotient
|
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