# On commutative rings whose prime ideals [6pt] are direct sums of cyclics

## M. Behboodi and A. Moradzadeh-Dehkordi

Address:

Corresponding author: M. Behboodi, Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box 84156-83111, Isfahan, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM) P.O.Box: 19395-5746, Tehran, Iran

Department of Mathematical Sciences, Isfahan University of Technology, P.O.Box 84156-83111, Isfahan, Iran

E-mail:

mbehbood@cc.iut.ac.ir

a.moradzadeh@math.iut.ac.ir

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 $(R, \mathcal{M})$, the following statements are equivalent: (1) Every prime ideal of $R$ is a direct sum of cyclic $R$-modules; (2) ${\mathcal{M}}=\bigoplus _{\lambda \in \Lambda }Rw_{\lambda }$ where $\Lambda $ is an index set and $R/{\operatorname{Ann}}(w_{\lambda })$ 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 where $\Lambda $ is an index set and ${\mathcal{M}}=\bigoplus _{\lambda \in \Lambda }Rw_{\lambda }$; 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 $(R, \mathcal{M})$ is a direct sum of (at most $n$) principal ideals, it suffices to test only the maximal ideal $\mathcal{M}$.

AMSclassification: primary 13C05; secondary 13E05, 13F10, 13E10, 13H99.

Keywords: prime ideals, cyclic modules, local rings, principal ideal rings.

DOI: 10.5817/AM2012-4-291