A Krull-Schmidt theorem for infinite sums of modules

with M' Sf, then there are submodules Di QD{ such that G = M'®(®i£iDi). Crawley and Jónsson proved ([3, Theorem 7.1] or [6, Theorem 7]) that if G is a direct sum of countably generated modules, each with the exchange property, then any two direct sum decompositions of G have isomorphic refinements. If G is also a direct sum of indecomposable modules, G = ffi ¿si Mi, then we can conclude that any direct sum decomposition of G refines into a decomposition isomorphic to this one, and, in particular, any summand of G is also isomorphic to a direct sum of indecomposable modules, each isomorphic to one of the Mi. In [3], this result is proved in the context of the theory of general algebraic systems, while in [ó] a version in Abelian categories is proved. Our first result characterizes the indecomposable modules which have the exchange property.