On Z-Symmetric Modules

A ring R is called a left Z-symmetric ring if ab ∈ Zₗ(R) implies ba ∈ Zₗ(R), where Zₗ(R) is the set of left zero-divisors. A right Z-symmetric ring is defined similarly, and a Z-symmetric ring is one that is both left and right Z-symmetric. In this paper, we introduce the concept of Z-symmetric modules as a generalization of Z-symmetric ring. Additionally, we introduce the concept of an eversible module as an analogy to eversible rings and prove that every eversible module is also a Z-symmetric module, like in the case of rings.