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

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∈ M and y∈ N such that every maximal submodule of M containing y also contains x, then x∈ N as well. We investigate the properties of z-submodules, particularly considering their stability with respect to various module constructions. Let Z(RM) denote the lattice of z-submodules of M ordered by inclusion. We are concerned with certain mappings between the lattices Z(RR) and Z(RM). The mappings in question are φ:Z(RR) → Z(RM) defined by setting for each z-ideal I of R, φ(I) to be the intersection of all z-submodules of M containing $IM$ and ψ:Z(RM) → Z(RR) defined by ψ(N) is the colon ideal $(N:M)$. It is shown that φ is a lattice homomorphism, and if M is a finitely generated multiplication module, then ψ is also a lattice homomorphism. In particular, Z(RM) is a homomorphic image of R(RM), 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 RY is a z-submodule of RY.