Let R be  commutative Ring , and let T be  unitary left .In this paper ,WAPP-quasi prime submodules are introduced as  new generalization of Weakly quasi prime submodules , where  proper submodule C of an R-module T is called WAPP –quasi prime submodule of T, if whenever 0≠rstϵC, for r, s ϵR , t ϵT, implies that either  r tϵ C +soc   or  s tϵC +soc  .Many examples of characterizations and basic properties are given . Furthermore several characterizations of WAPP-quasi prime submodules in the class of multiplication modules are established.

     We introduce in this paper, the notion of a 2-quasì-prime module as a generalization of quasi-prime module, we know that a module E over a ring R is called quasi-prime module, if (0) is quasi-prime submodule. Now, we say that a module E over ring R is a 2-quasi-prime module if (0) is 2-quasi-prime submodule, a proper submodule K of E is 2-quasi-prime submodule if whenever ,  and , then either  or .

Many results about these kinds of modules are obtained and proved, also, we will give a characterization of these kinds of modules.

     Let  be a commutative  ring with identity , and  be a unitary (left) R-module. A proper submodule  of  is said to be quasi- small prime submodule  , if whenever   with  and , then either or . In this paper ,we give a comprehensive study of quasi- small prime submodules.

        In this article, unless otherwise established, all rings are commutative with identity and all modules are unitary left R-module. We offer this concept of WN-prime as new generalization of weakly prime submodules. Some basic properties of weakly nearly prime submodules are given. Many characterizations, examples of this concept are stablished.

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. 

    Let R be a commutative ring with unity and let M, N be unitary R-modules. In this research, we give generalizations for the concepts: weakly relative injectivity, relative tightness and weakly injectivity of modules. We call M weakly N-quasi-injective, if for each f  Hom(N,) there exists a submodule X of  such that  f (N)  X ≈ M, where  is the quasi-injective hull of M. And we call M N-quasi-tight, if every quotient N / K of N which embeds in  embeds in M. While we call M weakly quasi-injective if M is weakly N-quasiinjective for every finitely generated R-module N.         Moreover, we generalize some properties of weakly N-injectiv

     Our aim in this work is to investigate prime submodules and prove some properties of them. We study the relations between prime submodules of a given module and the extension of prime submodules. The relations between prime submodules of two given modules and the prime submodules in the direct product of their quotient module are studied and investigated.

      In this paper, we introduce the notion of a 2-prime module as a generalization of prime module E over a ring R, where E is said to be prime module if (0) is a prime submodule. We introduced the concept of the 2-prime R-module. Module E is said to be 2-prime if (0) is 2-prime submodule of E. where a proper submodule K of module E is 2-prime submodule if, whenever rR, xE, E, Thus xK or [K: E].

    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