On Divided Modules

Abstract

Recall that a commutative ring R is said to be a divided ring if its each prime ideal P is comparable with each principal ideal (a), where $$a\in R.$$ In this paper, we extend the notion of divided rings to modules in two different ways: let R be a commutative ring with identity and M a unital R-module. Then M is said to be a divided (weakly divided) module if its each prime submodule N of M is comparable with each cyclic submodule Rm (rM) of M, where $$m\in M$$ ($$r\in R)$$. In addition to give many characterizations of divided modules, some topological properties of (quasi-) Zariski topology of divided modules are investigated. Also, we study the divided property of trivial extension $$R\ltimes M$$.