The aim of this paper is to introduces and study the concept of CSO-compact space via the notation of simply-open sets as well as to investigate their relationship to some well known classes of topological spaces and give some of his properties.

In the present paper, a simply* compact spaces was introduced it defined over simply*- open set previous knowledge and we study the relation between the simply* separation axioms and the compactness, in addition to introduce a new types of functions known as 𝛼𝑆 𝑀∗ _irresolte , 𝛼𝑆 𝑀∗ __𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 and 𝑅 𝑆 𝑀∗ _ continuous, which are defined between two topological spaces.

        Let R be a commutative ring with identity, and let M be a unitary R-module. We introduce a concept of almost bounded submodules as follows: A submodule N of an R-module M is called an almost bounded submodule if there exists xÃŽM, xÏN such that ann_R(N)=ann_R(x).

        In this paper, some properties of almost bounded submodules are given. Also, various basic results about almost bounded submodules are considered.

        Moreover, some relations between almost bounded submodules and other types of modules are considered.

A gamma T_ pure sub-module also the intersection property for gamma T_pure sub-modules have been studied in this action. Different descriptions and discuss some ownership, as Γ-module Z owns the TΓ_pure intersection property if and only if (J2 ΓK ∩ J^2  ΓF)=J^2 Γ(K ∩ F) for each Γ-ideal J and for all TΓ_pure K, and F in Z Q/P is TΓ_pure sub-module in Z/P, if P in Q.

In this paper, the concept of semi-?-open set will be used to define a new kind of strongly connectedness on a topological subspace namely "semi-?-connectedness". Moreover, we prove that semi-?-connectedness property is a topological property and give an example to show that semi-?-connectedness property is not a hereditary property. Also, we prove thate semi-?-irresolute image of a semi-?-connected space is a semi-?-connected space.

 Let R be a commutative ring with 10 and M is a unitary R-module . In this paper , our aim is to continue studying 2-absorbing submodules which are introduced by  A.Y. Darani and F. Soheilina . Many new properties and characterizations are given .

Let R be a commutative ring with unity and let M be a left R-module. We define a proper submodule N of M to be a weakly prime if whenever  r  R,  x  M, 0  r x  N implies  x  N  or  r  (N:M). In fact this concept is a generalization of the concept weakly  prime ideal, where a proper ideal P of R is called a weakly prime, if for all a, b  R, 0  a b  P implies a  P or b  P. Various properties of weakly prime submodules are considered.