On Direct Sum Decompositions of Krull-Schmidt Artinian Modules

TL;DR

This paper investigates the structure of Krull-Schmidt artinian modules, showing that their decompositions refine into finite indecomposables and exploring conditions for existence and uniqueness of such decompositions.

Contribution

It introduces Krull-Schmidt artinian modules and proves their decompositions refine into finite indecomposables, establishing a stronger condition than finite indecomposable decompositions.

Findings

Decompositions of Krull-Schmidt artinian modules refine into finite indecomposables

Krull-Schmidt artinian condition is stronger than finite indecomposable decompositions

Studied existence and uniqueness of decompositions in various module classes

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.