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  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.

  In this work we shall introduce the concept of weakly quasi-prime modules and give some properties of this type of modules.

"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 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 identity, and  be a unitary left -module. In this paper we introduce the concept pseudo weakly closed submodule as a generalization of -closed submodules, where a submodule  of an -module  is called a pseudo weakly closed submodule, if for all , there exists a -closed submodule  of  with  is a submodule of  such that . Several basic properties, examples and results of pseudo weakly closed submodules are given. Furthermore the behavior of pseudo weakly closed submodules in class of multiplication modules are studied. On the other hand modules with chain conditions on pseudo weakly closed submodules are established. Also, the relationships of  pseudo weakly closed

      Let R be a commutative ring with identity and M  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 end--prime if for each   EndR(M) and x  M, if (x)  P, then either x  P + (P) or (M)  P + (P). Some of the properties of this concept will be investigated. Some characterizations of end--prime submodules will be given, and we show that under some assumtions prime submodules and end--prime submodules are coincide.

Suppose R has been an identity-preserving commutative ring, and suppose V has been a legitimate submodule of R-module W. A submodule V has been J-Prime Occasionally as well as occasionally based on what’s needed, it has been acceptable: x ∈ V + J(W) according to some of that r ∈ R, x ∈ W and J(W) an interpretation of the Jacobson radical of W, which x ∈ V or r ∈ [V: W] = {s ∈ R; sW ⊆ V}. To that end, we investigate the notion of J-Prime submodules and characterize some of the attributes of has been classification of submodules.

In this research note approximately prime submodules is defined as a new generalization of prime submodules of unitary modules over a commutative ring with identity. A proper submodule  of an -module  is called an approximaitly prime submodule of  (for short app-prime submodule), if when ever , where , , implies that either  or . So, an ideal  of a ring  is called app-prime ideal of  if   is an app-prime submodule of -module . Several basic properties, characterizations and examples of approximaitly prime submodules were given. Furthermore, the definition of approximaitly prime radical of submodules of modules were introduced, and some of it is properties were established.