In this paper we used the concept of a pure submodule relative to submodule T in two concepts, pure relative to submodule T Baer injective modules and module with pure relative to submodule T intersection property. Some properties and some characterization of this notions are established.

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The aim of this work is studying many concepts of a pure submodule related to sub-module L and introducing the two concepts, T_pure submodule related to submodule and the crossing property of T_pure related to submodule. Another characterizations and study some properties of this concept.

Let R be a commutative ring with identity and M be a unitary R- module. We shall say that M is a primary multiplication module if every primary submodule of M is a multiplication submodule of M. Some of the properties of this concept will be investigated. The main results of this paper are, for modules M and N, we have M N and HomR (M, N) are primary multiplications R-modules under certain assumptions.

The main objective of this research is to study and to introduce a concept of strong fully stable Banach -algebra modules related to an ideal.. Some properties and characterizations of full stability are studied.