Completely simple endomorphism rings of modules
V. A. Bovdi M. A. Salim Mihail Ursul
math.RA math.GN 16W80, 16N20, 16S50, 16N40

Kinematics of CMEs and associated shock waves as deduced from LASCO data: comparative analysis
    Zagainova  Yu.S.
    
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It is proved that if A_p is a countable elementary abelian p-group, then:
(i) The ring End (A_p) does not admit a nondiscrete locally compact ring topology.
(ii) Under (CH) the simple ring  End (A_p)/I, where I is the ideal of End (A_p) consisting of all endomorphisms with finite images,  does not admit a nondiscrete locally compact ring topology.
(iii) The finite topology  on  End (A_p) is the only second metrizable ring topology on it.
Moreover, a characterization of completely simple endomorphism rings of the endomorphism rings of modules over  commutative rings  is also obtained.




§ INTRODUCTION

The notion of associative simple ring can be extended for associative  topological rings in several  ways:


  (i)
simple abstract ring endowed with a nondiscrete ring topology (for example,  the classification of nondiscrete locally compact division rings, see <cit.>, Chapter IV and  <cit.>; we refer to  some historical notes about  locally compact division rings to  <cit.>);

  (ii)
 topological ring without nontrivial closed ideals (for example,  see <cit.>).

  (iii)topological ring R with the property that if f:R→ S is a continuous homomorphism in a topological ring S, then either f=0 or f is a topological embedding of R into S (see <cit.>).


In all cases it is assumed that multiplication is not trivial.

I. Kaplansky has mentioned  (see <cit.>, p.56) that classification of locally compact simple rings in characteristic p>0  is difficult. He proved that every simple nondiscrete locally  compact simple torsion-free ring is a matrix ring over a locally  compact division  ring. However in <cit.> (see also <cit.>) has been constructed a nondiscrete locally  compact simple ring of positive characteristic which is not a  matrix ring over a division ring. Thereby the program of classification of nondiscrete locally compact simple rings was finished. Nevertheless it is interesting  to look for new examples of locally compact simple rings.

If A_p is a countable elementary abelian p-group and I is an ideal of  End (A_p) consisting of endomorphisms with finite images, then the factor ring End (A_p)/I is a simple von Neumann regular ring. We prove that under (CH) this ring does not admit a nondiscrete locally compact ring topology.



S. Ulam (see <cit.>, Problem 96, p.181)  posed  the following problem: "Can the group S_∞ of all permutations of integers so metrized that the group operation (composition of permutations) is a continuous function and the set S_∞ becomes, under this metric, a compact space? (locally compact?)". E.D. Gaughan (see <cit.>) has solved this problem in the negative.



In  <ref>   we  study an analogous problem for  the endomorphism ring of a countable elementary abelian p-group, namely: "Does the endomorphism ring End(A_p) of a countable elementary p-group A_p admit a nondiscrete locally compact ring topology?".  Similarly to Ulam's problem we obtain a negative answer to this question. Moreover, we prove that 𝒯_fin is the only ring topology 𝒯 on End (A_p) such that ( End (A_p), 𝒯) is complete and second metrizable.



We classify  in 4   the completely  simple rings ( End (M), 𝒯_fin)  of  vector spaces M over division rings.
Corollary <ref> gives a characterization of semisimple left linearly compact minimal rings.  It should be mentioned that Corollary <ref> is related to a result from <cit.> stating that any semisimple ring admits at most one linearly compact topology.



Furthermore,  we obtain in 5 a description of  completely simple rings of the form ( End (M_R), 𝒯_fin) of modules M over a commutative ring R. We extend the result of <cit.> to topological rings ( End (M_R), 𝒯_fin).










§ NOTATION, CONVENTIONS AND PRELIMINARY RESULTS


Rings are assumed to be associative, not necessarily with identity. Topological spaces are assumed to be completely regular.
The weight (see <cit.>, p.12) of the space X is denoted by w(X). The pseudocharacter of a point x∈ X  (see <cit.>, p.135) is the smallest cardinal of the form |𝒰|, where  𝒰  is a family of open subsets of X such that ∩𝒰={x}. The closure of a subset A of the topological space X is denoted by A and the interior by Int(A) (see <cit.>, p.14). A topological space X is called a Baire  space (see <cit.>, p.198) if for each sequence {X_1,X_2,…} of open dense subsets of X the intersection ∩_i=1^∞  G_i is a dense set.

An abelian group A is called elementary abelian p-group (p prime) if pa=0 for all a∈ A.  It is well-known  that   A  is a direct sum of copies of the cyclic group Z(p). The subring of a ring R generated by a subset S, is denoted by ⟨ S⟩. A ring R is called locally finite if every its finite subset is contained in a finite subring. A topological  ring (R,𝒯)  is called metrizable if its underlying additive group satisfies the first axiom of countability. A ring R with 1 is called Dedekind-finite if each equality xy=1 implies yx=1. It is well-known that every finite ring with identity is Dedekind-finite. Since every compact ring with
identity is a subdirect product of finite rings, it follows that every compact ring with identity is Dedekind-finite.
If A⊆  R, then  Ann_l(A):={x∈ R| xA=0}. If X,Y are the subsets of R, then X· Y:={xy| x∈ X, y∈ Y}.  A topological ring R is called compactly generated (see <cit.>, Chapter II) if there exists a compact subset K such that R=⟨ K⟩. If (R,𝒯) is a topological ring and I is an ideal of R, then the quotient topology of the factor ring R/I is denoted by 𝒯/I. If K is a  subset of  an abelian group A, then  set

    T(K)={α∈ End (A)|α(K)=0 }.

When K runs over all finite subsets of A, the family {T(K)} defines a ring topology 𝒯_fin on
End (A). This topology is called the finite topology.


 
For any abelian group A the ring ( End   A,𝒯_fin) is complete.


See <cit.>, Theorem 19.2.



(Cauchy's criterion) In a Hausdorff complete commutative group G, in order that a family (x_α)_α∈Ω should be summable it is necessary and sufficient that, for each neighborhood V of zero in G, there is a finite subset Ω_0 of Ω such that Σ_α∈ Kx_α∈ V for all finite subsets K of Ω which do not meet Ω.


See <cit.>, p.263.




If (x_α)_α∈Ω is a summable subset in ( End   A,𝒯_fin) then  every subset Δ of Ω the family (x_β)_β∈Δ is summable.


Let V be a neighborhood of zero of ( End   A,𝒯_fin). We can consider without loss of generality that V is a left ideal of End   A. There exists a finite subset Ω_0 of Ω such that Σ_α∈ Kx_α∈ V for every finite subset K of Ω for which K∩Ω_0=∅. Let F be a finite subset of Δ such that F∩(Ω_0∩Δ)=∅. If α∈ F, then α∉Ω_0, hence Σ_α∈ Fx_α∈ V. By Cauchy's criterion the family (x_β)_β∈Δ is summable.



A topological ring (R,𝒯) is called minimal (see, for instance, <cit.>)  if there is no ring topology 𝒰 such that 𝒰≤𝒯
and 𝒰≠𝒯. A topological ring (R,𝒯) is called simple if  R is simple as a ring without topology. A topological ring (R,𝒯) is called weakly simple  if  R^2≠ 0 and  every its closed ideal is either  0 or R.
A topological ring (R,𝒯) is called completely simple (see  <cit.>)  if  R^2≠ 0 and  for every continuous homomorphism   f: (R,𝒯)→(S,𝒰) in a topological ring (S,𝒰) either  (f)=R or f is a homeomorphism of (R,𝒯) on Im(f). Equivalently,   R^2≠ 0 and   (R,𝒯) is weakly simple and minimal.
Let M be a unitary right R-module over   a commutative ring R  with 1.
The  module M is called divisible if Mr=M for every 0≠ r∈ R.
A  right  R-module M is called faithful if Mr=0 implies r=0 (r∈ R).
A   right  R-module M is called torsion-free if mr=0 implies that either m=0 or r=0, where m∈ M and r∈ R.
Recall that a submodule N of an R-module M is called fully invariant α(N)⊆ N for every endomorphism α of M_R.
We use in the sequel the notion and results from  the books <cit.>.



If R is a von Neumann regular ring, then R^2=R.



An ideal I of a von Neumann regular ring is von Neumann regular.



Let i∈ I. Thus there exists x∈ R such that ixi=i. It follows that ixixi=i and xix∈ I.


If I an ideal of  a von Neumann regular ring R, then any ideal H of I is an ideal of R, too.


RH=RH^2⊆ IH⊆ H. Similarly, HR⊆ H.


If A_p is a p-elementary countable group, then

    I={α∈ End (A_p)|| Im(α)|<ℵ_0}.

Fix a linear basis {v_i| i∈ℕ} of A_p over the field 𝔽_p. Using this fixed  basis, we define  the map  e_i: A → A such that

    e_i(v_j)=δ_ijv_j,      (i, j∈ℕ)

where δ_ij is the Kronecker delta.

 We have for End (A_p):


  (i) I is a von Neumann regular ring.

  (ii) I is a simple ring.

  (iii) The factor ring End(A_p)/I is  simple  von Neumann regular.

  (iv) I is a locally finite ring.




(i):  It is well-known that  End(A_p) is regular (see <cit.>, Theorem 4.27, p.63). Then  I is von Neumann regular by Lemma <ref>.

(ii), (iii):  The ideal  I is the only nontrivial ideal of   End(A_p) (see <cit.>, 17, Theorem 1,  p.93).  This means that  End(A)/I  is simple. It is  regular by the part (i).

(iv)  Since I is simple (see <cit.>,  12, Proposition 1),  it suffices to show that I contains a nonzero locally finite right ideal.

Let us show  that the left ideal Ie_1 of I is locally finite as a ring  (equivalently, as a 𝔽_p-algebra). We have 0≠ e_1∈ Ie_1.  If H is the left annihilator of Ie_1, then, obviously, H is a locally finite ring, hence it is locally finite as a 𝔽_p-algebra. We claim that Ie_1/H is finite.
Indeed, define β_n∈ H (n≥ 2) in the following way

    β_n(v_i)=
     v_n,  for   i=1;
    
    0,    for   i≠ 1.

We claim that Ie_1=𝔽_pe_1+Σ_n=2^∞𝔽_pβ_n.

For, let α∈ I. Then α(v_1)=r_1v_1+⋯+r_nv_n, where r_i∈𝔽_p and n∈ℕ.  Thus

    α e_1(v_1)    =r_1e_1(v_1)+r_2β_2(v_1)+⋯+r_nβ_n(v_1)
       =(r_1e_1+r_2β_2+⋯+r_nβ_n)(v_1);
    α e_1(v_j)   =(r_1e_1+r_2β_2+⋯+r_nβ_n)(v_j)     (j≠ 1).

This yields

    α e_1=r_1e_1+r_2β_2+⋯+r_nβ_n

and so Ie_1=𝔽_pe_1+Σ_n=2^∞𝔽_pβ_n.

In particular,
Ie_1=𝔽_pe_1+H, and so H has a finite index in Ie_1. Clearly,  Ie_1 is a locally finite 𝔽_p-algebra (see <cit.>,  Proposition 1,  p.241) and  I is a locally finite 𝔽_p-algebra (see <cit.>, Proposition 2, p.242).



The next result  can be deduced from Lemma 36.11 of <cit.>.

Let A be a  locally compact, compactly generated, and  totally disconnected ring. If  A contains a dense   locally finite subring B, then A is compact.


Let A=⟨ V⟩, where V is a compact symmetric neighborhood of zero. Since V is compact, the subset V+V+V· V also is compact. Since B is dense, A=B+V. By compactness of V+V+V· V there exists a 	 finite subset H⊆  B such that V+V+V· V⊆ H+V. Since B is a locally finite ring, we can assume without loss of generality that H is a subring. Let  H∖{0}={h_1,…,h_k}.
The subset

    H+h_1V+⋯+h_kV+V

 is an open subgroup of R(+). Indeed, this subset is symmetric and

    (H+h_1V+⋯+h_kV+V)+(H+h_1V+⋯+h_kV+V)   
    ⊆ H+h_1(V+V)+⋯+h_k(V+V)+V+V   
    ⊆ H+h_1V+⋯+h_kV+V   .

We  prove by induction on m that

    V^[m]⊆ H+h_1V+⋯+h_kV+V,       (m∈ℕ)

where V^[1]=V and V^[m]=V^[m-1]· V for all m.

The inclusion is obvious for m=1.

Assume that the assertion has been proved for m≥1. Clearly,

    V^[m+1]=V^[m]· V⊆ H· V+h_1(V· V)+⋯ + h_k(V· V)+V· V   ⊆
    
    h_1V+⋯+ h_kV+h_1(H+V)+⋯+ h_k(H+V)+H+V   ⊆
    
    H+h_1V+⋯+h_kV+V   .

Consequently,   A=H+h_1V+⋯+h_kV+V, therefore A is compact.



An element x of a topological ring is called discrete if there exists a neighborhood V of zero such that xV=0 (i.e., the right annihilator of x is open).



The set of all discrete elements of  a topological ring is  an ideal. A simple ring with identity does not contain nonzero discrete elements.




§ LOCALLY COMPACT RING TOPOLOGIES  ON  END (A) OF A COUNTABLE ELEMENTARY ABELIAN P-GROUP A







Let R be a simple,   nondiscrete and  locally compact ring of  char(R)=p>0 and 1∈ R.
If V is a compact open subring of R and {e_α|α∈Ω} is a set of orthogonal idempotents in R, then

    |Ω|≤ w(V).



The ring R does not contain nonzero discrete elements by Lemma  <ref>. Since R is locally compact and char(R)=p,  it is totally disconnected. Additionally, R has a fundamental system of neighborhoods of zero consisting of compact open subrings by  Lemma 9 of <cit.>.

If V is a compact open subring of R, then by continuity of the ring operations  for each α∈Ω there exists an open ideal V_α of V such that e_α V_α⊆ V.  Clearly, there exists   y_α∈ V_α for which   e_α y_α≠ 0 since R has no nonzero discrete elements.

We claim that hold the following two properties:

(i) e_α y_α∉{e_β y_β|β≠α} for each α∈Ω;

(ii) the set    X={e_α y_α|α∈Ω} is a discrete subspace of V.

Indeed, if were  e_α y_α∈{e_β y_β|β≠α} for some α∈Ω, then

    e_α y_α=e_α e_α y_α   ∈ e_α{e_β y_β|β≠α}
       ⊆{e_α e_β y_β|β≠α}
       ={0},

so  e_α y_α=0, a contradiction. The part (i) is proved.

(ii) Now,   for  each α∈Ω  we have  V∖{e_β y_β|β≠α} is open and, consequently,

    (V∖{e_β y_β|β≠α})∩ X={e_α y_α},

by (i). Therefore the point e_α y_α(α∈Ω) of X is isolated. In other words, the subspace X of V is discrete.



Since X is discrete,      |Ω|=|X|=w(X)≤ w(V)  (see <cit.>, Exercises 98-99, p.72).




Let A_p be a countable elementary abelian p-group.
Then the ring 
    I={α∈ End (A_p)|| Im(α)|<ℵ_0}

 does not admit a nondiscrete  ring topology 𝒰
such that (I,𝒰) is a Baire space.



Put S_n={α∈ I|α(A)⊆𝔽_p v_1+⋯+𝔽_pv_n}, where n∈ℕ.
Clearly  I=∪_n∈ℕS_n and

    S_n={α∈ I | e_iα=0   for    i> n}= Ann_r({e_k| k>n}).

This yields that   S_n is closed due continuity of the ring operations.


Since I is a Baire space, there exists n∈ℕ such that Int(S_n)≠∅, hence S_n is an open subgroup.

Set β∈ I such that

    β(v_i)=
     v_n+i,  for   i=1,…,n;
    
    0  ,    for   i> n.

Let W⊆ S_n be a neighborhood of zero of (I,𝒰) such that β W⊆ S_n. If w∈ W∖ 0,  then there exist a∈ A and r_1,…, r_n∈𝔽_p such that

    0≠ w(a)=∑_i=1^n r_iv_i    and    β( w(a))=∑_i=1^n  r_iv_n+i.

There exists  j∈ 1,…, n such that r_j≠ 0. Then

    e_n+jβ w(a)=r_jv_n+j≠ 0,

 hence e_n+jβ w≠ 0 and so  β w∉S_n, a contradiction.



Under the notation of Theorem <ref> the ring  I does not admit a nondiscrete locally compact ring topology.


This follows from the fact that  each  locally compact space is a Baire space  (see <cit.>, Theorem 1, p.117).


Our main result is the following.


The endomorphism ring End (A_p) of a countable elementary abelian p-group A_p does not admit a nondiscrete locally compact ring topology.



We will use the notation and results from section 2.
Denote R=  End (A_p). Assume on the contrary that there exists on R a nondiscrete locally compact ring topology 𝒯.

Fact 1. The ring (R,𝒯) has a fundamental system of neighborhoods of zero consisting of compact open subrings.


Since  the additive group of R  has exponent p, it  is totally
disconnected (this follows from <cit.>, Theorem 9.14, p.95).
By   I. Kaplansky's result (see <cit.>, Lemma 9) (R,𝒯) has a  fundamental system of neighborhoods of zero consisting of compact open subrings.

Fact 2. The group  Re_i
is countable for each i∈ℕ.

We claim  that Re_i is infinite. Indeed, for each
j∈ℕ  put β_j∈ R such that

    β_j(v_k)=
    
    v_j,   for   k=i;
    
    0,     for    k≠i.

If j≠s, then β_je_i(v_i)=β_j(v_i)=v_j and
β_se_i(v_i)=β_s(v_i)=v_s, hence β_je_i≠β_se_i, so Re_i
is infinite.

The ring Re_i is countable. Indeed, consider the mapping ψ: Re_i→ A_p^𝔽_pv_i, where

    ψ(α e_i)(rv_i)=α(rv_i)    for all      r∈𝔽_p.

If α e_i≠β e_i (α, β∈ R), then there exists x=∑_j r_jv_j∈ A_p such that
α e_i(x)≠β e_i(x), hence,  α (r_iv_i)≠β(r_iv_i). Thus

    ψ(α e_i)(r_iv_i)=α (r_iv_i)≠β(r_iv_i)=ψ(β e_i)(r_iv_i).

The latter means that ψ is an injective mapping of  Re_i into A^𝔽_pv_i .
Since A^𝔽_pv_i is countable, Re_i is countable, too.

Fact 3. I is a closed ideal of R.
 We claim that I is not dense in  (R,𝒯).
Assume the contrary.
Since I  is locally  finite and is a maximal ideal,   (R,𝒯)
is topologically locally finite by Lemma  <ref>. The ring R contains two elements
x, y such that xy=1 and yx≠ 1. The subring ⟨ x,y⟩
is compact, hence Dedekind-finite, a contradiction. We obtained that
(R/I,𝒯/I) is a nondiscrete  metrizable locally compact ring.


Fact 4. I is a discrete ideal of R.

This follows from Theorem 2.

Fact  5. Re_i is a discrete left ideal of R for every i∈ℕ.

Indeed, Re_i⊆ I and I is discrete by Fact 4 for every i∈ℕ.

Fact  6.  Ann_l(e_i) is  open in R for every i∈ℕ.

Indeed, the group homomorphism
q: R→ Re_i, r↦ re_i, is continuous. Since Re_i is discrete q^-1(0)=  Ann_l(e_i)
is open.

Fact 7.   ∩_i Ann_l(e_i)=0.

Obvious.

Fact 8. 𝒯≥𝒯_fin.

We notice that Ann_l(e_i)=T({v_i}) for every i∈ℕ. For, if α e_i=0,  then α(v_i)=α e_i(v_i)=0, i.e., α∈ T({v_i}).
Conversely, if  α∈ T({v_i}), then α e_i(v_i)=α(v_i)=0. If j≠i then α e_i(v_j)=0. Therefore
α e_i=0. Furthermore

    T({v_1,…,v_n})=∩_i=1^n T({v_i})=∩_i=1^n  Ann_l(e_i)∈𝒯
 for each n∈ℕ.

Since the family  {T({v_1,…,v_n})} forms a fundamental system of  neighborhoods
of zero of (R,𝒯_fin), we get that 𝒯_fin≤𝒯.

Fact 9. The ring (R,𝒯)  is metrizable.

Since ∩_i∈ℕ Ann_l(e_i)=0, the pseudocharacter of (R,𝒯) is ℵ_0.
If V is a compact open subring of (R,𝒯) (see Fact 1), then the pseudocharacter of V also is ℵ_0.
However in every compact space the pseudocharacter of a point coincides with its character. Therefore (R,𝒯)  is metrizable.

Fact 10.
(R/I,𝒯/I) has an open compact subring.

Indeed, it is well-known (see <cit.>) that every totally disconnected ring has a fundamental system of
neighborhood of zero consisting of compact open subrings.
Henceforth  V is  a fixed open compact  subring of (R/I,𝒯/I).

Fact 11.  R/I contains a family of  orthogonal idempotents of cardinality 2^ℵ_0 .

Indeed, the family {e_i}_i∈ℕ of idempotents of  (R, 𝒯_fin) is summable in  and 1_A=Σ_n∈ℕe_n, where 1_A is the identity of R.

The first ordinal  number of cardinality 𝔠 of continuum is denoted by ω(𝔠). Let {ℕ(α)|α< ω(𝔠)}  be a family of infinite almost disjoint subsets of ℕ (see <cit.>, Example 3.6.18, p.175–176).
Put  f_ℕ(α)=Σ_i∈ℕ(α)e_i for every α<ω(𝔠). The element f_ℕ(α) exists by Lemma <ref>. Then:


  (i) f_ℕ(α)∉ I for every α  <ω(𝔠);

  (ii) f_ℕ(α)f_ℕ(β)∈ I for each α,β <ω(𝔠) and α≠β.

If  g_α=f_ℕ(α)+I for each α< ω(𝔠),  then {g_α|α <ω(𝔠)} is the required system of orthogonal idempotents.

The subring  V is metrizable (by Fact 9). Since V is  compact and R/I is a simple von Neumann regular  ring by Lemma  <ref> and  w(V)≤ℵ_0,  we obtain a   contradiction to Theorem <ref>. 


(CH)
Under the notation of Theorem <ref>,  the ring R/I does not admit a nondiscrete locally compact ring topology.


Assume on the contrary that  the factor ring R/I admits  a nondiscrete locally compact ring topology 𝒯, so (R/I, 𝒯) contains an open compact subring V. Since the cardinality of R/I is continuum and V is infinite, the power of V is continuum. Since we have assumed (CH), the subring  V is  metrizable, hence second metrizable (see <cit.>).  However  we have proved in  Theorem <ref> that the ring  R/I contains a family of orthogonal idempotents of cardinality 𝔠, a  contradiction with Theorem <ref>.


The finite topology 𝒯_fin is the only second metrizable ring topology 𝒯 on R  for which (R,𝒯_fin) is complete.


Let K=⟨ F⟩, where  F is  a finite subset of A. Clearly,  there exists a subgroup A^' of A such that A=K⊕ A^'.
Choose  e_F∈ R such that e_F↾_K= id_K and e_F(A^')=0.  Clearly

    T(K)=R(1-e_F)

and   α K=0 if and only if  α∈ R(1-e_F). It follows that the family
{R(1-e_F)} where  F runs over all finite subset of A, forms a fundamental system
of neighborhoods of zero for (R,𝒯_fin).

There exists an injective map of Re_F to Hom(K,A), so the left ideal Re_F
is countable, due to countability  Hom(K,A). Since e_F^2=e_F, the Peirce decomposition

    R=Re_F⊕ R(1-e_F)

of R with respect to the idempotent e_F is a decomposition of the  topological group (R,+,𝒯).  It follows that Re_F is discrete, hence R(1-e_F) is open (in the topology 𝒯). It follows that 𝒯≥𝒯_fin, so 𝒯=𝒯_fin (see <cit.>,  Theorem 30    or <cit.>). 



§ COMPLETELY SIMPLE TOPOLOGICAL ENDOMORPHISM RINGS
 OF VECTOR SPACES




Let A_F be a right vector space over a division ring F and S= End (A_F).
The following conditions are equivalent:


  (i) (S, 𝒯_fin) is a completely simple
topological ring.

  (ii) (A_F)=∞ or (A_F)< ∞ and F does not admit a nondiscrete ring topology.



(i)⇒ (ii): If A_F is finite-dimensional, then S is discrete and isomorphic to
M(n,F), where n is the dimension of A_F. Then, obviously, F does not admit a
nondiscrete ring topology.



(ii)⇒ (i):
If (A_F)= n< ∞, then  S≅ M(n,F). Since F does not admit nondiscrete ring topologies, the same holds for  M(n,F).

Let A_F be  infinite dimensional. Fix a   basis {x_α}_α<τ over F, where τ
is an infinite ordinal number. It is well-known  that the topological ring   (S, 𝒯_fin) is weakly  simple (see <cit.>, Satz 12, p.258) and  the family {T(x_α)}_α<τ is a prebase at
zero for the finite topology 𝒯_fin of S.


Assume on the contrary that there exists a Hausdorff ring topology 𝒯,
coarser that 𝒯_fin and different from  it. Let e_α∈ S such that e_α^2=e_α and  e_α(x_β)=δ_αβx_α for each α<τ, where  δ_αβ is the Kronecker delta.


Fact 1. T(x_α)= Ann_l(e_α) for each α<τ.


Indeed, if p∈ T(x_α), then pe_α(x_α)=p(x_α)=0. If β≠α, then e_α(x_β)=0, hence pe_α=0, i.e. p∈ Ann_l(e_α).
Conversely, if pe_α=0, then p(x_α)=pe_α(x_α)=0, i.e.  p∈ T(x_α).



Fact 2. There exists α_0<τ for which Se_α_0 is nondiscrete in (S,𝒯).

Assume on the contrary that   for every α<τ there exists a neighborhood V_α of zero of (S,𝒯) such that Se_α∩ V_α=0. If U_α is a neighborhood of zero of  (S,𝒯) such that U_α e_α⊆ V_α, then U_α e_α=0, hence Ann_l(e_α)=T(x_α) is open in (S,𝒯). Hence  𝒯_fin≤𝒯 and  𝒯=𝒯_fin, a contradiction.

Fact 3.   (Se_α_0∩ V)x_α_0⊈⊕_β∈ Kx_β F

for any neighborhood V of of zero of (S,𝒯) and any finite subset K of the set [0,τ) of all ordinal numbers less than τ.


Assume on the contrary that there exists a finite subset K of  [0,τ) and a neighborhood V of zero of (S,𝒯) such that

    (Se_α_0∩ V)x_α_0⊆⊕_β∈ Kx_β F.

Fix γ∈ [0,τ)∖ K. For each β∈ K define  q_β∈ S such that q_β(x_β)=x_γ and q(x_δ)=0 for δ≠β.

Let V_0 be a neighborhood of zero of (S,𝒯) such that V_0⊆ V and q_β V_0⊆ V for all β∈ K.
There exists 0≠ h∈ Se_α_0∩ V_0 by  Fact 2. It follows that hx_α_0≠0 by  Fact 1.
Since Se_α_0∩ V_0⊆ Se_α_0∩ V, we obtain that  hx_α_0=Σ_β∈ Kx_β f_β, (f_β∈ F)  by (<ref>). There exists β_0∈ K such that f_β_0≠ 0 (because hx_α_0≠0), so

    q_β_0h=q_β_0(Σ_β∈ Kx_β f_β)=r_β_0x_γ∉⊕_β∈ Kx_β F,

a contradiction. Therefore  Fact 3 is proved.

Now let V be a  neighborhood of zero of (S,𝒯). Pick up    a neighborhood V_0 of zero of (S,𝒯)  such that V_0· V_0⊆ V.
Since 𝒯≤𝒯_fin,   there exists a finite subset K of [0,τ) such that

    T({x_β|β∈ K})⊆ V_0.

We have   (Se_α_0∩ V_0)x_α_0⊈⊕_β∈ Kx_β F by  Fact 3. It follows that there exists  q∈ Se_α_0∩ V_0 such that

    q(x_α_0)∉⊕_β∈ Kx_β F.

Clearly,  q(x_α_0)∈ A_F, so it  can be written as  q(x_α_0)=∑_α< τx_αf_α, where f_α∈ F and here exists β_0∉K  such that f_β_0≠0.

Consider the element   s∈ S such that  s(x_β_0)=x_α_0f_β_0^-1 and s(x_λ)=0 for λ≠β_0. Evidently, s∈ T(K), hence

    sq∈ T(K)· V_0⊆ V_0·  V_0⊆ V.

Moreover, sq(x_α_0)=s(x_β_0f_β_0+⋯)=x_α_0. Since q∈ Se_α_0 we obtain that sq(x_β)=0 for β≠α_0. Consequently,  e_α_0=sq∈ V for every neighborhood V of zero of (S,𝒯), a contradiction.


The question of existence of a uncountable division ring which does not admit a nondiscrete Hausdorff ring topology is open.
Several results on this topic  can be found in Chapter 5 of <cit.>.



Let  ∏_α∈ΩR_α be a family of compact rings with identity. Then the product (∏_α∈ΩR_α,∏_α∈Ω𝒯_α) is a minimal ring if and only if  every (R_α,𝒯_α) is a minimal topological ring. (Here ∏_α∈Ω𝒯_α is the product topology on ∏_α∈ΩR_α.)


⇒: Assume on the contrary that there exists β∈Ω and a ring topology 𝒯^' on R_β such that 𝒯^'≤𝒯_β and 𝒯^'≠𝒯_β. Consider the product topology 𝒰 on ∏_α∈ΩR_α, where R_α is endowed with 𝒯_α when α≠β and R_β is endowed with 𝒯^'. Obviously, 𝒰≤∏_α∈Ω𝒯_α and 𝒰≠∏_α∈Ω𝒯_α, a contradiction.

⇐:
Denote by π_α(α∈Ω) the projection of ∏_α∈ΩR_α on R_α.  By definition of the product topology, ∏_α∈Ω𝒯_α is the coarsest topology on ∏_α∈ΩR_α for which the projections π_α(α∈Ω) are continuous.

Let 𝒰  be a ring topology on  ∏_α∈ΩR_α, 𝒰≤∏_α∈Ω𝒯_α and β∈Ω. Since

    𝒰↾_R_β×∏_γ≠β{0_γ}≤(∏_α∈Ω𝒯_α)↾_R_β×∏_γ≠β{0_γ},

it follows that 𝒰↾_R_β×∏_γ≠β{0_γ}=(∏_α∈Ω𝒯_α)↾_R_β×∏_γ≠β{0_γ} by minimality of (R_β,𝒯_β).

Then the family {V×∏_γ≠β{0_γ}} when V runs all neighborhoods of zero of (R_β,𝒯_β) is  a fundamental system of neighborhoods of zero of

    (R_β×∏_γ≠β{0_γ},  𝒰↾_R_β×∏_γ≠β{0_γ}).

Since R_β×∏_γ≠β{0_γ} is an ideal with identity of ∏_α∈ΩR_α, the topological ring (∏_α∈ΩR_α,𝒰) is a direct sum of ideals R_β×∏_γ≠β{0_γ} and {0_β}×∏_γ≠βR_γ.
Let  V be a neighborhood of zero of (R_β,𝒯_β).
Then V×∏_γ≠βR_γ will be a neighborhood of zero of (∏_α∈ΩR_α,𝒰) and π_β(V×∏_γ≠βR_γ)=V.

We have proved that π_β is a continuous function from (∏_α∈ΩR_α,𝒰) to (R_β,𝒯_β). It follows that  ∏_α∈Ω𝒯_α≤𝒰 and so 𝒰=∏_α∈Ω𝒯_α.







A left linearly compact semisimple ring is minimal if and only if has no direct summands of the form M(n,Δ), where Δ is a division ring which does not admit a nondiscrete Hausdorff ring topology.


This follows from Theorems <ref>, <ref> and the Theorem of Leptin (see <cit.>, Theorem 13, p. 258) about the structure of  left linearly compact semisimple rings.




A semisimple linearly compact ring (R, 𝒯) having no ideals isomorphic to matrix rings over infinite division rings is minimal.





§ COMPLETELY SIMPLE ENDOMORPHISM RINGS OF MODULES


The  endomorphism ring of a right R-module M is denoted by

 End (M_R).


Let M be  a  divisible,  torsion-free module over a commutative domain R and  K  the  field of fractions of R. The additive group of  M has a structure of   a vector K-space such that R-endomorphisms of M are exactly the  K-linear transformations.


We define a structure of a right vector K-space as follows: if  a/b∈ K and m∈ M, then there exists a unique x∈ M such that ma=xb; set m∘a/b =x.
Moreover,  if  a/b=c/d and 0≠ m∈ M, then m ∘a/b=m ∘c/d. Indeed, if m ∘a/b=x and m ∘c/d=y, then mad=xbd and mbc=ybd which means that xbd=ybd, hence x=y.

Let α∈ End (M_R), a/b∈ K, m∈ M. By definition, am=b(a/b∘ m), hence, aα(m)= bα(a/b∘ m), which means that α(a/b∘ m)=a/b∘α(m), so α is a K-linear transformation.
Note that,  if a∈ R and m∈ M, then m ∘a/1=ma.

Conversely, if α is a K-linear transformation, a∈ R, m∈ M, then
α(a/1∘ m)=a/1∘α m, i.e. α(am)=aα(m). We have proved that every K-linear transformation is an right R-module homomorphism. 


The center Z(R) of a weakly simple ring  R is a domain.


For every right R-module M the underlying group M(+) is a discrete left topological   ( End (M_R),𝒯_fin)-module.

Indeed, T(m)(m)=0 for every m∈ M. Moreover,

    End (M_R){0}={0},

 so M is a discrete left topological   ( End (M_R),𝒯_fin)-module.




Let M_R be a module over a commutative ring R.

If the topological ring ( End (M_R),𝒯_fin) is weakly simple, then:


  (i) P={r∈ R| Mr=0} is a prime ideal of R.


  (ii) M is a vector space over the field K of fractions of  R/P and  the R-endomorphisms of M are exactly the  K-linear transformations.



Conversely,  if M_R is an R-module and are satisfied (i) and (ii), then  ( End (M_R),𝒯_fin) is a weakly simple topological ring.



⇒:
If  ( End (M_R),𝒯_fin) is weakly simple, then  the mapping:

    α_r: M → M,       m↦ mr      (r∈ R)

is an R-module homomorphism and α_r∈ Z(= the center of End (M_R)).

First we show that the part (i) holds.  Indeed, if a,b∈ R and ab=0, then α_aα_b=0 (see (<ref>)). Thus ( End (M_R)α_a)· ( End (M_R)α_b)=0, so

    ( End (M_R)α_a)· ( End (M_R)α_b)=0.

Since End (M_R) is weakly simple, one of them,  say  End (M_R)α_a, is zero.  This implies that α_a=0, hence a∈ P.


(ii) The structure of R/P-module  on M is defined as follows:  if r∈ R and m∈ M, then put M(r+P)=mr.

Note  that M is a torsion-free right R/P-module. Assume  that  m(r+P)=0, where   0≠ r+P∈ R/P and  0≠ m∈ M. Then mr=0=α_r(m) (see (<ref>)).  Thus End (M_R)α_r(m)=0. It follows that  ( End (M_R)α_r)(m)=0 by  Remark <ref>. Since End (M_R) is weakly simple

    End (M_R)α_r= End (M_R).

We obtained that End (M_R)(m)=0, so m=0, a contradiction.


 Under this convention  R-submodules are exactly R/P-submodules and R-endomorphisms are exactly
R/P-endomorphisms.

The module  M is a divisible R/P-module. Indeed, if  0≠ r+P∈ R/P, then  0≠M(r+P)=Mr. Suppose that Mr≠ M. Consider

    I={α∈ E|α(M)⊆ Mr}.

Since Mr is a fully invariant submodule, I is a two-sided ideal of ( End (M_R),𝒯_fin).

The ideal I is closed. Indeed, let α∈I. If m∈ M, then there exists β∈ I such that α-β∈ T(m). Clearly,   α(m)=β(m)∈ Mr and so α∈ I. We have proved that I is closed.



Since 1_M∉ I, I=0. It follows that    α_r=0 (see (<ref>)),  a contradiction.



The module M has a structure of a right K-vector space and End (M_R) is exactly the ring of endomorphisms of M by Lemma <ref>.


The converse follows from  Theorem <ref>.





A characterization of completely simple topological ring End (M_R) is given by the following.


 Let M_R be a module over a commutative ring R.
The topological  ring ( End (M_R), 𝒯_fin) is completely simple if and only  are satisfied the conditions (i) and (ii) of Theorem 8 and either


  (i) M is finite or

  (ii) M is infinite and the dimension of M over the field K  is infinite.





⇒: According to Theorem <ref>, the ideal P is prime   and the topology of  End (M_R) coincide with the finite topology of  End (M_K), where K is the field of fractions of R/P. If M is finite, we have the part (i).
Assume that M is infinite. If R/P is finite, then the dimension of M over K is infinite. Suppose that R/P is infinite and _K(M)=n<ℵ_0. Then M is isomorphic to M(n,K). Since K is an infinite field, it admits a nondiscrete ring topology (see <cit.>) and we obtain a contradiction because End (M_R) is a discrete ring. Consequently  _K(M) is infinite.


⇐ This follows from Theorems <ref> and <ref>.



The topological ring ( End (A), 𝒯_fin) of an abelian group A is completely  simple if and only  one of the following conditions holds:


  (i) A is a elementary abelian  p-group.

  (ii) A is a divisible torsion-free group of infinite rank.






abbrv

