The main goal of this paper is to introduce a new class in the category of modules. It is called quasi-invertibility monoform (briefly QI-monoform) modules. This class of modules is a generalization of monoform modules. Various properties and another characterization of QI-monoform modules are investigated. So, we prove that an R-module M is QI-monoform if and only if for each non-zero homomorphism f:M E(M), the kernel of this homomorphism is not quasi-invertible submodule of M. Moreover, the cases under which the QI-monoform module can be monoform are discussed. The relationships between QI-monoform and other related concepts such as semisimple, injective and multiplication modules are studied. We also show that they are proper subclass

he concept of small monoform module was introduced by Hadi and Marhun, where a module U is called small monoform if for each non-zero submodule V of U and for every non-zero homomorphism f ∈ Hom R (V, U), implies that ker f is small submodule of V. In this paper the author dualizes this concept; she calls it co-small monoform module. Many fundamental properties of co-small monoform module are given. Partial characterization of co-small monoform module is established. Also, the author dualizes the concept of small quasi-Dedekind modules which given by Hadi and Ghawi. She show that co-small monoform is contained properly in the class of the dual of small quasi-Dedekind modules. Furthermore, some subclasses of co-small monoform are investiga

In this paper, we introduce and study the notions of fuzzy quotient module, fuzzy (simple, semisimple) module and fuzzy maximal submodule. Also, we give many basic properties about these notions.

Throughout this note, R is commutative ring with identity and M is a unitary R-module. In this paper, we introduce the concept of quasi J-  submodules as a     –  and give some of its basic properties. Using this concept, we define the class of quasi J-regular modules, where an R-module     J- module if every submodule of  is quasi J-pure. Many results about this concept

In this paper, we introduce a type of modules, namely S-K-nonsingular modules, which is a generalization of K-nonsingular modules. A comprehensive study of these classes of modules is given.

A Module M is called cofinite  J- Supplemented  Module  if for every  cofinite submodule L of  M, there exists a submodule N of M such that M=L+N with   main properties of cof-J-supplemented modules.  An R-module M is called fully invariant-J-supplemented if for every fully invariant submodule N of M, there exists a submodule K of M, such that M = N + K with N  K K. A condition under which the direct sum of FI-J-supplemented modules is FI-J-supplemented was given. Also, some types of modules that are related to the FI-J-supplemented module were discussed.

     In this work, we introduce a new generalization of both Rationally extending and Goldie extending which is Goldie Rationally extending module which is known as follows: if for any submodule K of an R-module M there is a direct summand U of M (denoted by  U⊆_⊕ M) such that K β_r  U. A β_r  is a relation of K⊆M and U⊆M, which defined as  K β_r  U if and only if  K ⋂U⊆_r K and K⋂U⊆_r U.

We introduce in this paper the concept of an approximately pure submodule as a     generalization of a pure submodule, that is defined by Anderson and Fuller. If every submodule of an R-module  is approximately pure, then  is called F-approximately regular. Further, many results about this concept are given.

       A submodule N of a module M  is said to be s-essential if it has nonzero intersection with any nonzero small submodule in M. In this article, we introduce and study a class of modules in which all its nonzero endomorphisms have non-s-essential kernels, named, strongly -nonsigular. We investigate some properties of strongly -nonsigular modules. Direct summand, direct sums and some connections of such modules are discussed.        

      Fuchs introduced purely extending modules as a generalization of extending modules. Ahmed and Abbas gave another generalization for extending modules named semi-extending modules. In this paper, two generalizations of the extending modules are combined to give another generalization. This generalization is said to be almost semi-extending. In fact, the purely extending modules lies between the extending and almost semi-extending modules. We also show that an almost semi-extending module is a proper generalization of purely extending. In addition, various examples and important properties of this class of modules are given and considered. Another characterization of almost semi-extending modules is established. Moreover, the re