Abstract
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.
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