Krull-Schmidt uniqueness fails dramatically over subrings of Z\(\oplus\dots \oplus Z\)
One version of the Krull-Schmidt-Azumaya theorem states that if Z is a complete commutative Noetherian local ring, R is a Z-algebra which is finitely generated as a Z-module and M is a finitely generated left R- module, then M is a finite sum of indecomposable R-modules and this sum is unique up to isomorphism and the order of occurrence of the summands. It is well-known [see for example \textit{C. W. Curtis} and \textit{I. Reiner}, ''Methods of representation theory, with applications to finite groups and orders. I'' (1981; Zbl 0469.20001); Section 36] that the theorem does not hold if Z is not required to be complete. This paper exhibits spectacular failure of the K-S-A theorem for some concrete rings and modules. Indeed, let Z be the integers; \(p_ 2,...,p_ m\), \(q_ 2,...,q_ m\) be distinct prime numbers; and R be the Z-submodule of \(\oplus^{m}Z\) generated by \((1,1,...,1),p_ 2q_ 2(0,1,1,...,1),\quad p_ 3q_ 3(0,0,1,1,...,1),...,p_ mq_ m(0,...,0,1).\)Endow R with coordinate- wise multiplication. For suitable choices of p's and q's: there is a module M such that M can be expressed as the direct sum of k indecomposable modules for every k with 2\(\leq k\leq m\); there are modules M,N,C with \(M\oplus C\cong N\oplus C\) but M not isomorphic to N; there are modules M and N with \(\oplus^{n}M\cong \oplus^{n}N\) but M not isomorphic to N; there are four non-isomorphic indecomposable modules H,K,M,N such that \(H\oplus K\cong M\oplus N\), and two maximal ideals P,Q of R such that \(H_ P\cong M_ P\neq K_ P\cong N_ P\), but \(H_ Q\cong N_ Q\neq K_ Q\cong M_ Q.\)
decomposition into indecomposable modules
scientific article; zbMATH DE number 3857252