On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics

TL;DR

This paper characterizes commutative rings with prime ideals as direct sums of cyclic modules, especially focusing on finite products of local rings and providing criteria for Noetherian local rings.

Contribution

It provides a complete structural description of such rings and establishes a practical criterion for checking prime ideals in Noetherian local rings.

Findings

Prime ideals in certain rings are direct sums of cyclic modules.

Characterization of local rings where prime ideals are sums of cyclic modules.

A criterion reduces checking prime ideals to the maximal ideal in Noetherian local rings.

Abstract

In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring $(R, \cal{M})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) ${\cal{M}}=\bigoplus_{\lambda\in \Lambda}Rw_{\lambda}$ and $R/{\rm Ann}(w_{\lambda})$ is a principal ideal ring for each $\lambda \in \Lambda$;(3) Every prime ideal of $R$ is a direct sum of at most $|\Lambda|$ cyclic $R$-modules; and (4) Every prime ideal of $R$ is a summand of a direct sum of cyclic $R$-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring $(R, \cal{M})$ is a direct sum of (at most $n$) principal ideals, it suffices to test only the maximal ideal $\cal{M}$.