Commutative rings whose proper ideals are direct sum of cyclically presented modules

TL;DR

This paper investigates commutative rings where every ideal can be expressed as a direct sum of cyclically presented modules, extending the understanding of ideal structure in ring theory.

Contribution

It introduces a study of commutative rings with ideals decomposable into cyclically presented modules, expanding on classical results about principal ideals.

Findings

Characterization of rings with ideals as direct sums of cyclically presented modules

Conditions under which ideals decompose into cyclically presented modules

Connections to classical results on principal ideals

Abstract

A famous result due to I. M. Isaacs states that if a commutative ring $R$ has the property that every prime ideal is principal, then every ideal of $R$ is principal. This motivates ring theorists to study commutative rings for which every ideal is a direct sum of cyclically presented modules. In this paper, we study commutative rings whose ideals are direct sum of cyclically presented modules.