Commutative Local Rings whose Ideals are Direct Sums of Cyclic Modules

TL;DR

This paper investigates commutative local rings where every ideal decomposes into a direct sum of cyclic modules, extending previous work by removing the Noetherian restriction.

Contribution

It generalizes prior results by studying non-Noetherian commutative local rings with ideals as direct sums of cyclic modules.

Findings

Characterization of such rings without Noetherian assumption

Extension of known results to broader class of rings

New structural insights into ideals as direct sums of cyclic modules

Abstract

A well-known result of K\"{o}the and Cohen-Kaplansky states that a commutative ring $R$ has the property that every $R$-module is a direct sum of cyclic modules if and only if $R$ is an Artinian principal ideal ring. This motivated us to study commutative rings for which every ideal is a direct sum of cyclic modules. Recently, in [M. Behboodi, A. Ghorbani, A. Moradzadeh-Dehkordi, Commutative Noetherian local rings whose ideals are direct sums of cyclic modules, J. Algebra 345 (2011) 257--265] the authors considered this question in the context of finite direct products of commutative Noetherian local rings. In this paper, we continue their study by dropping the Noetherian condition.