On a generalization of $z$-ideals in modules over commutative rings

Abstract

In this article, we introduce and study the concept of $z$-submodules as a generalization of $z$-ideals. Let $M$ be a module over a commutative ring with identity $R$. A proper submodule $N$ of $M$ is called a $z$-submodule if for any $x\in M$ and $y\in N$ such that every maximal submodule of $M$ containing $y$ also contains $x$, then $x\in N$ as well. We investigate the properties of $z$-submodules, particularly considering their stability with respect to various module constructions. Let $\mathcal{Z}({_R}M)$ denote the lattice of $z$-submodules of $M$ ordered by inclusion. We are concerned with certain mappings between the lattices $\mathcal{Z}({_R}R)$ and $\mathcal{Z}({_R}M)$. The mappings in question are $\phi:\mathcal{Z}({_R}R) \rightarrow \mathcal{Z}({_R}M)$ defined by setting for each $z$-ideal $I$ of $R$, $\phi(I)$ to be the intersection of all $z$-submodules of $M$ containing $IM$ and $\psi:\mathcal{Z}({_R}M) \rightarrow \mathcal{Z}({_R}R)$ defined by $\psi(N)$ is the colon ideal $(N:M)$. It is shown that $\phi$ is a lattice homomorphism, and if $M$ is a finitely generated multiplication module, then $\psi$ is also a lattice homomorphism. In particular, $\mathcal{Z}({_R}M)$ is a homomorphic image of $\mathcal{R}({_R}M)$, the lattice of radical submodules of $M$. Finally, we show that if $Y$ is a finite subset of a compact Hausdorff $P$-space $X$, then every submodule of the $C(X)$- module $\mathbb{R}^Y$ is a $z$-submodule of $\mathbb{R}^Y$.