Commutative Noetherian and Krull Rings by Stanislaw Balcerzyk, Tadeusz Jozefiak

By Stanislaw Balcerzyk, Tadeusz Jozefiak

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The ring K[X1 , .. 6 If 0 ~ M' ~ M .! M" ~ 0 is an exact sequence of R-modules, then the module Mis Noetherian if and only if the modules M' and M" are Noetherian. Proof Assume that M is Noetherian. 3. Similarly, every submodule of M" is a homomorphic image of a submodule of M, whence it is finitely generated. Noetherian Rings and Modules 56 [Ch. Conversely, suppose that M' and M" are Noetherian and let N be an arbitrary submodule of M. (M')flN and {J(N) are finitely generated. (M')flN, and let x 1 , ..

Is by definition an exact sequence if and only if Ker({") = Im(f"- 1 ) for all n. In particular, a sequence 0 -> M' ~ M .!.. M" -> 0 is an exact sequence if and only if oc is a monomorphism, {J is an epimorphism, and Ker({J) = Im(o:). Such a sequence is called a short exact sequence. Every element me M determines a cyclic submodule, {rm; re R}, written Rm. Similarly, for every ideal I in R and every element m e M, we denote by Im the set {rm; re/}, which is a submodule of the module M. For any subset A c: M, the smallest submodule of M containing A consists of all elements of the form_r 1 a 1 + ...

0 = {f e C(X); f(xo) = 0}. for some point x 0 e X. Show that the correspondence x phism between the spaces X and Max(C(X)). 1-+ m,. determines a homeomor- Preliminary Concepts 28 [Ch. 3 MODULES Let R be a ring. An R-module is by definition a triplet (M, +, ·)where M is a non-empty set, + : M x M -+ M, · : Rx M -+ M, the following conditions. being satisfied for all elements r, s e R and m 1 , m2 e M: (i) M is an Abelian group with respect to the operation +, (ii) r(m 1 +m2) = rm1 +rm2, (iii) (r+s)m 1 = rm1 +sm1, (iv) r(sm 1) = (rs)m1, (v) lm1 = m1.

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