In this paper, we introduce the notation of the soft bornological group to solve the problem of boundedness for the soft group. We combine soft set theory with bornology space to produce a new structure which is called    soft bornological group. So that both the product and inverse maps are soft bounded. As well as, we study the actions of the soft bornological group on the soft bornological sets. The aim soft bornological set is to partition into orbital classes by acting soft bornological group on the soft bornological set. In addition, we explain the centralizer, normalizer, and stabilizer in details. The main important results are to prove that the product of soft bornological groups is soft bornol

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 the concept of a quasi-radical semi prime submodule. Throughout this work, we assume that    is a commutative ring with identity and  is a left unitary R- module. A  proper submodule  of  is called a quasi-radical semi prime submodule (for short Q-rad-semiprime), if     for   ,   ,and then  . Where   is the intersection of all prime submodules of .

Let M be an R-module. In this paper we introduce the concept of quasi-fully cancellation modules as a generalization of fully cancellation modules. We give the basic properties, several characterizations about this concept. Also, the direct sum and the localization of quasi-fully cancellation modules are studied.

     In this paper, we define and study z-small quasi-Dedekind as a generalization of small quasi-Dedekind modules. A submodule  of -module  is called z-small (  if whenever  , then . Also,  is called a z-small quasi-Dedekind module if for all  implies  . We also describe some of their properties and characterizations. Finally, some examples are given.

  Let R be a commutative ring with unity. In this paper we introduce and study the concept of strongly essentially quasi-Dedekind module as a generalization of essentially quasiDedekind module. A unitary R-module M is called a strongly essentially quasi-Dedekind module if ( , ) 0 Hom M N M for all semiessential submodules N of M. Where a submodule N  of  an R-module  M  is called semiessential if , 0  pN for all nonzero prime submodules  P of  M .
 

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

In this research, we introduce a small essentially quasi−Dedekind R-module to generalize the term of an essentially quasi.−Dedekind R-module. We also give some of the basic properties and a number of examples that illustrate these properties.