$z^{\circ }$-submodules of a reduced multiplication module

Abstract

Let R be a commutative ring with identity and M be an R-module. A proper ideal I of R is said to be a $z^{\circ }$-ideal if for each $a\in I$ the intersection of all minimal prime ideals containing a is contained in I. The purpose of this paper is to introduce the notion of $z^{\circ }$-submodules of M as an extension of $z^{\circ }$-ideals of R. Moreover, we investigate some properties of this class of submodules when M is a reduced multiplication R-module.