In this paper we study the concepts of copure submodules and coregular
modules. Many results related with these concepts are obtained.

    In this paper, we introduce the concept of s.p-semisimple module. Let S be a semiradical property, we say that a module M is s.p - semisimple if for every submodule N of M, there exists a direct summand K of M such that K ≤ N and N / K has S. we prove that a module M is s.p - semisimple module if and only if for every submodule A of M, there exists a direct summand B of M such that A = B + C and C has S. Also, we prove that for a module M is s.p - semisimple if and only if for every submodule A of M, there exists an idempotent e ∊ End(M) such that e(M) ≤ A and (1- e)(A) has S. 

Let  R be an associative ring with identity and let M be a left R-module . As a generalization of µ-semiregular modules, we introduce an F-µ-semiregular module. Let F be a submodule of M and x∊M. x is called F-µ-semiregular element in M , if there exists a decomposition M=A⨁B, such that A is a projective submodule of  and . M is called  F-µ-semiregular if x is F-µ-semiregular element for each x∊M. A condition under which the module µ-semiregular is F-µ-semiregular module was given. The basic properties and some characterizations of the F-µ-semiregular module were provided.

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.

The concept of fully pseudo stable Banach Algebra-module (Banach A-module) which is the generalization of fully stable Banach A-module has been introduced. In this paper we study some properties of fully stable Banach A-module and another characterization of fully pseudo stable Banach A-module has been given.

     The new type of paranormal operators that have been defined in this study on the Hilbert space, is  paranormal operators. In this paper we introduce and discuss some properties of this concept such as: the sum and product of two paranormal, the power of paranormal. Further, the relationships between the paranormal operators and other kinds of paranormal operators have been studied.

Let m ≥ 1,n ≥ 1 be fixed integers and let R be a prime ring with char (R) ≠2 and
(m+n). Let T be a (m,n)(U,R)-Centralizer where U is a Jordan ideal of R and T(R)
⊆ Z(R) where Z(R) is the center of R ,then T is (U,R)- Centralizer.

The main goal of this paper is to introduce and study a new concept named d^*-supplemented which can be considered as a generalization of W- supplemented modules and d-hollow module. Also, we introduce a d^*-supplement submodule. Many relationships of d^*-supplemented modules are studied. Especially, we give characterizations of d^*-supplemented modules and relationship between this kind of modules and other kind modules for example every d-hollow (d-local) module is d^*-supplemented and by an example we show that the converse is not true.

Let R be an associative ring with identity, and let M be a unital left R-module, M is called totally generalized *cofinitely supplemented module for short ( T G*CS), if every submodule of M is a Generalized *cofinitely supplemented ( G*CS ). In this paper we prove among the results under certain condition the factor module of T G*CS is T G*CS and the finite sum of T G*CS is T G*CS.

In this paper the full stable Banach gamma-algebra modules, fully stable Banach gamma-algebra modules relative to ideal are introduced. Some properties and characterizations of these classes of full stability are studied.