Notes on approximately pure submodules

Abstract EN

Let R be a commutative ring with identity 1 and M be a unitary left R-module.

A submodule N of an R-module M is said to be approximately pure submodule of an R-module, if for each ideal I of R.

The main purpose of this paper is to study the properties of the following concepts: approximately pure essentialsubmodules, approximately pure closedsubmodules and relative approximately pure complement submodules.

We prove that: when an R-module M is an approximately purely extending modules and N be Ap-puresubmodulein M, if M has the Ap-pure intersection property then N is Ap purely extending.