Our aim in this paper is to study the relationships between min-cs modules and some other known generalizations of cs-modules such as ECS-modules, P-extending modules and n-extending modules. Also we introduce and study the relationships between direct sum of mic-cs modules and mc-injectivity.

This paper introduces the concept of fuzzy σ-ring as a generalization of fuzzy σ-algebra and basic properties; examples of this concept have been given.  As the first result, it has been proved that every  σ-algebra over a fuzzy set x*  is a  fuzzy σ-ring-over a fuzzy set x*  and construct their converse by example. Furthermore, the fuzzy ring  concept has been studied to generalize fuzzy algebra and its relation. Investigating  that the concept of fuzzy  σ-Ring is a stronger form of a fuzzy ring  that is every fuzzy σ-Ring over a fuzzy set x* is a fuzzy ring over a fuzzy set x* and construct their converse by example. In addition, the idea of the smallest, as an important property in the study of real analysis, is studied

A non-zero submodule N of M is called essential if N L for each non-zero submodule L of M. And a non-zero submodule K of M is called semi-essential if K P for each non-zero prime submodule P of M. In this paper we investigate a class of submodules that lies between essential submodules and semi-essential submodules, we call these class of submodules weak essential submodules.

  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 .
 

Let R be a commutative ring with identity and let M be a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of semi-essential submodules which introduced by Ali S. Mijbass and Nada K. Abdullah, and we make simple changes to the definition relate with the zero submodule, so we say that a submodule N of an R-module M is called semi-essential, if whenever N ∩ P = (0), then P = (0) for each prime submodule P of M. Various properties of semi-essential submodules are considered.

      Let  be a right module over a ring  with identity. The semisecond submodules are studied in this paper. A nonzero submodule  of   is called semisecond if    for each . More information and characterizations about this concept is provided in our work.

  In this paper, we introduce and study the concept of S-coprime submodules, where a proper submodule N of an R-module M is called S-coprime submodule if M N is S-coprime Rmodule. Many properties about this concept are investigated.

      Sequences spaces  , m  ,  p  have called quasi-Sobolev spaces were  introduced   by Jawad . K. Al-Delfi in 2013  [1]. In this  paper , we deal with notion of  quasi-inner product  space  by using concept of  quasi-normed  space which is generalized  to normed space and given a  relationship  between  pre-Hilbert space and a  quasi-inner product space with important  results   and   examples.  Completeness properties in quasi-inner   product space gives  us  concept of  quasi-Hilbert space .  We show  that ,  not  all  quasi-Sobolev spa