Let  be a commutative ring with an identity and be a unitary -module. We say that a non-zero submodule  of  is  primary if for each with en either or  and an -module  is a small primary if   =  for each proper submodule  small in. We provided and demonstrated some of the characterizations and features of these types of submodules (modules).  

    Suppose that A be an abelain ring with identity, B be a unitary (left) A-module, in this paper ,we introduce a type of modules ,namely Quasi-semiprime A-module, whenever   is a Prime Ideal For proper submodule N of  B,then B is called Quasi-semiprime module ,which is a Generalization of Quasi-Prime A-module,whenever  ann_AN is a prime ideal for proper submodule N of B,then B is Quasi-prime module .A comprchensive study of these modules is given,and we study the Relationship between quasi-semiprime module and quasi-prime .We put the codition coprime over cosemiprime ring for the two cocept quasi-prime module and quasi-semiprime module are equavelant.and the cocept of  prime module and quasi

The present study introduces the concept of J-pure submodules as a generalization of pure submodules. We  study some of its basic  properties  and  by using this concept we  define the class of  J-regular modules,  where an R-module  M is called  J-regular module if every submodule of M is J-pure submodule. Many results about this concept are proved

In this paper, we introduce the concept of e-small Projective modules as a generlization of Projective modules.

     This paper aims to introduce the concepts of  -closed, -coclosed, and -extending modules as generalizations of the closed, coclossed, and extending modules,  respectively. We will prove some properties as when the image of the e*-closed submodule is also e*-closed and when the submodule of the e*-extending module is e*-extending. Under isomorphism, the e*-extending modules are closed. We will study the quotient of e*-closed and e*-extending, the direct sum of e*-closed, and the direct sum of e*-extending.

Throughout this paper, T is a ring with identity and F is a unitary left module over T. This paper study the relation between semihollow-lifting modules and semiprojective covers. proposition 5 shows that If T is semihollow-lifting, then every semilocal T-module has semiprojective cover. Also, give a condition under which a quotient of a semihollow-lifting module having a semiprojective cover. proposition 2 shows that if K is a projective module. K is semihollow-lifting if and only if For every submodule A of K with K/( A) is hollow, then K/( A) has a semiprojective cover.

An R-module M is called a 2-regular module if every submodule N of M is 2-pure submodule, where a submodule N of M is 2-pure in M if for every ideal I of R, I2MN = I2N, [1]. This paper is a continuation of [1]. We give some conditions to characterize this class of modules, also many relationships with other related concepts are introduced.

Let  be a commutative ring with 1 and  be a left unitary . In this paper, the generalizations for the notions of compressible module and  retractable module are given.

An   is termed to be  semi-essentially compressible if   can be embedded in every of a non-zero semi-essential submodules. An  is termed a semi-essentially retractable module, if   for every non-zero semi-essentially submodule of an . Some of their advantages characterizations and examples are given.  We also study the relation between these classes and some other classes of modules.

he concept of small monoform module was introduced by Hadi and Marhun, where a module U is called small monoform if for each non-zero submodule V of U and for every non-zero homomorphism f ∈ Hom R (V, U), implies that ker f is small submodule of V. In this paper the author dualizes this concept; she calls it co-small monoform module. Many fundamental properties of co-small monoform module are given. Partial characterization of co-small monoform module is established. Also, the author dualizes the concept of small quasi-Dedekind modules which given by Hadi and Ghawi. She show that co-small monoform is contained properly in the class of the dual of small quasi-Dedekind modules. Furthermore, some subclasses of co-small monoform are investiga

      Fuchs introduced purely extending modules as a generalization of extending modules. Ahmed and Abbas gave another generalization for extending modules named semi-extending modules. In this paper, two generalizations of the extending modules are combined to give another generalization. This generalization is said to be almost semi-extending. In fact, the purely extending modules lies between the extending and almost semi-extending modules. We also show that an almost semi-extending module is a proper generalization of purely extending. In addition, various examples and important properties of this class of modules are given and considered. Another characterization of almost semi-extending modules is established. Moreover, the re