Let  be a commutative ring with identity, and a fixed ideal of  and  be an unitary -module. In this paper we  introduce and study the concept of -nearly prime submodules as genrealizations of nearly prime and we investigate some properties of this class of submodules. Also, some characterizations of -nearly prime submodules will be given.

           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.

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. 

"In this article, "we introduce the concept of a WE-Prime submodule", as a stronger form of a weakly prime submodule". "And as a "generalization of WE-Prime submodule", we introduce the concept of WE-Semi-Prime submodule, which is also a stronger form of a weakly semi-prime submodule". "Various basic properties of these two concepts are discussed. Furthermore, the relationships between "WE-Prime submodules and weakly prime submodules" and studied". "On the other hand the relation between "WE-Prime submodules and WE-Semi-Prime submodules" are consider". "Also" the relation of "WE-Sime-Prime submodules and weakly semi-prime submodules" are explained. Behind that, some characterizations of these concepts are investigated".

    Let  be a non-zero right module over a ring  with identity. The weakly second submodules is studied in this paper. A non-zero submodule  of   is weakly second Submodule when  ,  where ,  and  is a submodule of  implies either  or   . Some connections between these modules and other related modules are investigated and number of conclusions  and characterizations are gained.

Our aim in this paper is to introduce the notation of nearly primary-2-absorbing submodule as generalization of 2-absorbing submodule where a proper submodule  of an -module  is called nearly primary-2-absorbing submodule if whenever , for , , ,  implies that either  or  or . We got many basic, properties, examples and characterizations of this concept. Furthermore, characterizations of nearly primary-2-absorbing submodules in some classes of modules were inserted. Moreover, the behavior of nearly primary-2-absorbing submodule under -epimorphism was studied.

      Let R be a commutative ring with identity and M be an unitary R-module. Let (M) be the set of all submodules of M, and : (M)  (M)  {} be a function. We say that a proper submodule P of M is -prime if for each r  R and x  M, if rx  P, then either x  P + (P) or r M  P + (P) . Some of the properties of this concept will be investigated. Some characterizations of -prime submodules will be given, and we show that under some assumptions prime submodules and -prime submodules are coincide. 

Let be a commutative ring with unity and let be a submodule of anon zero left R-module , is called semiprime if whenever , implies . In this paper we say that is nearly semiprime, if whenever , implies ( ),(in short ),where ( )is the Jacobson radical of . We give many results of this type of submodules.