  We study direct sum decompositions of modules satisfying the descending chain
condition on direct summands. We call modules satisfying this condition
Krull-Schmidt artinian. We prove that all direct sum decompositions of
Krull-Schmidt artinian modules refine into finite indecomposable direct sum
decompositions and we prove that this condition is strictly stronger than the
condition of a module admitting finite indecomposable direct sum
decompositions. We also study the problem of existence and uniqueness of direct
sum decompositions of Krull-Schmidt artinian modules in terms of given classes
of modules. We present also brief studies of direct sum decompositions of
modules with deviation on direct summands and of modules with finite
Krull-Schmidt length.
