prime ideal of A under extension is one of the central problems of algebraic number theory. The following is sometimes useful: a prime ideal is a contraction of a prime ideal if and only if . (Proof: Assuming the latter, note intersects , a contradiction. Now, the prime ideals of correspond to those in B that are disjoint from . Hence, there is a prime ideal of B, disjoint from , such that is a maximal ideal containing . One then checks that lies over . The converse is obvious.) Generalizations Ideals can be generalized to any monoid object , where is the object where the monoid structure has been forgotten. A left ideal of is a subobject that "absorbs multiplication from the left by elements of "; that is, is a left ideal if it satisfies the following two conditions: is a subobject of For every and every , the product is in . A right ideal is defined with the condition "" replaced by "'". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone. An ideal can also be thought of as a specific type of -module. If we consider as a left -module (by left multiplication), then a left ideal is really just a left sub-module of . In other words, is a left (right) ideal of if and only if it is a left (right) -module which is a subset of . is a two-sided ideal if it is a sub--bimodule of . Example: If we let , an ideal of is an abelian group which is a subset of , i.e. for some . So these give all the ideals of . See also Modular arithmetic Noether isomorphism theorem Boolean prime ideal theorem Ideal theory Ideal (order theory) Ideal norm Splitting of prime ideals in Galois extensions Ideal sheaf Notes References Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Perseus Books, 1969, Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. External links Algebraic structures Commutative algebra Algebraic number theory