Unique decomposition and isomorphic refinement theorems in additive categories

Abstract

Under suitable hypotheses, if an object in an additive category is a direct sum of subobjects with local endomorphism rings, then any two such decompositions have the summands isomorphic in pairs, and any other decomposition has a refinement into a decomposition where the summands are of this sort. The purpose of this paper is to give careful proofs of such results, using what appear to be minimal hypotheses. Theorems of this sort for infinite direct sums in additive, non-Abelian categories, have been essential in recent work in Abelian g;oup theory. In 1909, MacLagan-Wedderburn [29] proved that if a finite group is expanded in two ways as a direct sum of indecomposable factors, then the summands are isomorphic in pairs. Remak [ 181 showed that the summands are actually centrally isomorphic. This theorem was extended by Gull [ 131 and Schmidt [ 19,201 to operator groups satisfying the double chain condition for admissible subgroups, and the numerous extensions of this famous theorem are most often referred to as Krull-Schmidt Theorems. In this paper we are primarily interested in generalizations of such theorems concerning infinite direct sums of modules. The most famous such result is Azymaya’s unique decomposition theorem, published in 1950: Let M be a module which has a direct sum deconlposition us a finite or infinite direct stun of irzdecomposable subrmdules Mi (i E I), such that the endornorphisrn ring oj’each Mi (i E I) is a local ritzg. Therz any imieconzposable sutmzand oj’M is isomorphic to Mi Ibr some i E I, md if M is the direct sum of the iudecotnposable submodules NjQ’ E J), therz there is a bijective mapping Q, : I + J such that Mi s NQco for all i E I. h-r 1964, Crawley and Jonsson [6] published some unique decomposition and isomorphic refinement theorein for general algebraic systems. A key condition in their work was’the so-called exchange property (defined in Section 2 below). In