Weak Krull–Schmidt for Infinite Direct Sums of Uniserial Modules

Abstract

w x The well-known Krull]Schmidt]Azumaya theorem 2 says that if M s [ M s [ N are two indecomposable decompositions of a i j ig I jg J module M, where M has local endomorphism ring for each i g I, then i there is a bijection s : I a J such that M ( N for every i g I. In this i s Ž i. theorem the local endomorphism ring hypothesis is crucial to ensure the uniqueness of the decomposition of M into indecomposable summands, and it is always tempting to ask whether under some weaker hypotheses the indecomposable decomposition of a module is unique. This was done, w x for example, by Anderson and Fuller 1, Theorem 12.4 , who showed that the conclusion of the Krull]Schmidt]Azumaya theorem still remains valid for modules with an indecomposable decomposition that complements maximal direct summands. In general, however, there exist various examples in the literature which show that Krull]Schmidt may fail even under ‘‘rather good’’ hypotheses on the rings or on the indecomposable sumŽ w x w x w x. mands see, e.g., 5 , 7 , and 8 . In 1975 Warfield proved that every finitely presented module over a serial ring is a finite direct sum of uniserial modules, and asked whether