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: Z (RR) Z (RM) defined by setting for each z-ideal I of R, (I) to be the intersection of all z-submodules of M containing IM and: Z (RM) Z (RR) defined by (N) is the colon ideal (N: M). It is shown that is a lattice homomorphism, and if M is a finitely generated multiplication module, then 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.