Some results on pure submodules relative to submodule

Abstract EN

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

A submodule N of an R-module M is said to be pure relative to submodule T of M (Simply T-pure) if for each ideal A of R, ( ) In this paper, the properties of the following concepts were studied: Pure essential submodules relative to submodule T of M (Simply T-pure essential),Pure closed submodules relative to submodule T of M (Simply T-pure closed) and relative pure complement submodule relative to submodule T of M (Simply T-pure complement) and T-purely extending.

We prove that; Let M be a T-purely extending module and let N be a T-pure submodule of M.

If M has the T-PIP, then N is T-purely extending.