In this article, we introduce a class of modules that is analogous of generalized extending modules. First  we define a module M to be a generalized ECS if and only if for each ec-closed submodule A of M, there exists a direct summand D of M such that  is singular, and then we locate generalized ECS between the other extending generalizations. After that we present some of characterizations of generalized ECS condition. Finally, we show that the direct sum of a generalized ECS need not be generalized ECS and deal with decompositions for be generalized ECS concept.

In this notion we consider a generalization of the notion of a projective modules , defined using y-closed submodules . We show that for a module M = M1M2 . If M2 is M1 – y-closed projective , then for every y-closed submodule N of M with M = M1 + N , there exists a submodule M`of N such that M = M1M`.

     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 ring with 1 and D is a left module over . In this paper, we study the relationship between essentially small quasi-Dedekind modules with scalar and multiplication modules. We show that if D is a scalar small quasi-prime -module, thus D is an essentially small quasi-Dedekind -module. We also show that if D is a faithful multiplication -module, then D is an essentially small prime -module iff is an essentially small quasi-Dedekind ring.

 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 associative; ring; with an identity and let D be unitary left R- module; . In this work we present semiannihilator; supplement submodule as a generalization of R-a- supplement submodule, Let U and V be submodules of an R-module D if D=U+V and whenever Y≤ V and D=U+Y, then annY≪R;. We also introduce the the concept of semiannihilator -supplemented ;modules and semiannihilator weak; supplemented modules, and we give some basic properties of this conseptes

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

   Let R be a commutative ring with unity and let M be a unitary R-module. In this paper we study fully semiprime submodules and fully semiprime modules, where a proper fully invariant R-submodule W of M is called fully semiprime in M if whenever XXW for all fully invariant R-submodule X of M, implies XW.         M is called fully semiprime if (0) is a fully semiprime submodule of M. We give basic properties of these concepts. Also we study the relationships between fully semiprime submodules (modules) and other related submodules (modules) respectively.

The purpose of this paper is to investigate the concept of relative quasi-invertible submodules motivated by rational submodules and quasi-invertible submodules. We introduce several properties and characterizations to relative quasi-invertiblity. We further investigate conditions under which identification consider between rationality, essentiality and relative quasi-invertiblity. Finally, we consider quasiinvertiblity relative to certain classes of submodules

In this paper, we introduce a class of operators on a Hilbert space namely quasi-posinormal operators that contain properly the classes of normal operator, hyponormal operators, M–hyponormal operators, dominant operators and posinormal operators . We study some basic properties of these operators .Also we are looking at the relationship between invertibility operator and quasi-posinormal operator .