On indecomposable decompositions of CS-modules

It is shown that, over any ring R, the direct sum M = (+i is an element of l)M(i) of uniform right R-modules M(i) with local endomorphism rings is a CS-module if and only if every uniform submodule of M is essential in a direct summand of M and there does not exist an infinite sequence of non-isomorphic monomorphisms [GRAPHICS] with distinct i(n) is an element of I. As a consequence, any CS-module which is a direct sum of submodules with local endomorphism rings has the exchange property.