Let R be a commutative ring with non-zero identity element. For two fixed positive integers m and n. A right R-module M is called fully (m,n) -stable relative to ideal A of , if for each n-generated submodule of Mm and R-homomorphism . In this paper we give some characterization theorems and properties of fully (m,n) -stable modules relative to an ideal A of . which generalize the results of fully stable modules relative to an ideal A of R.

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.

The aim of this paper is introducing the concept of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal. Some properties of (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal have been studied and another characterizations have been given. The relationship of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal that states,  a B- -module Ӽ is (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal  , if and only if  for any two ɱ-element sub-sets and of Ӽɳ, if , for each j = 1, …, ɱ,  i = 1,…, ɳ  and   implies Ạɳ( ) Ạɳ(  have been proved..

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.

MOHAMED ALIIt is shown that the classification of polynomial algebras over the mod p Steenrod algebra is an essentially different problem from the classification of polynomial algebras truncated at height greater than p over the Steenrod algebra .