On Commutative Rings Whose Prime Ideals Are Direct Sums of Cyclics

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 $\cal{M}$.