The main objective of this research is to study and to introduce a concept of strong fully stable Banach -algebra modules related to an ideal.. Some properties and characterizations of full stability are studied.

        Let R be a commutative ring with  identity . In this paper  we study  the concepts of  essentially quasi-invertible submodules and essentially  quasi-Dedekind modules  as  a generalization of  quasi-invertible submodules and quasi-Dedekind  modules  . Among the results that we obtain is the following : M  is an essentially  quasi-Dedekind  module if and only if M is aK-nonsingular module,where a module M is K-nonsingular if, for each  , Kerf ≤_e M   implies   f = 0 .

Let be an associative ring with identity and let be a unitary left -module. Let  be a non-zero submodule of .We say that  is a semi- - hollow module if for every submodule  of  such that  is a semi- - small submodule ( ). In addition, we say that  is a semi- - lifting module if for every submodule  of , there exists a direct summand  of  and  such that  

The main purpose of this work was to develop the properties of these classes of module.

In this article, we study the notion of closed Rickart modules. A right R-module M is said to be closed Rickart if, for each , is a closed submodule of M. Closed Rickart modules is a proper generalization of Rickart modules. Many properties of closed Rickart modules are investigated. Also, we provide some characterizations of closed Rickart modules. A necessary and sufficient condition is provided to ensure that this property is preserved under direct sums. Several connections between closed Rickart modules and other classes of modules are given. It is shown that every closed Rickart module is -nonsingular module. Examples which delineate this concept and some results are provided.

Let R be a commutative ring with unity 1 6= 0, and let M be a unitary left module over R. In this paper we introduce the notion of epiform∗ modules. Various properties of this class of modules are given and some relationships between these modules and other related modules are introduced.

      In this paper, we introduced module that satisfying strongly -condition modules and strongly -type modules as generalizations of t-extending. A module  is said strongly -condition if for every submodule of  has a complement which is fully invariant direct summand. A module   is said to be strongly -type modules if every t-closed submodule has a complement which is a fully invariant direct summand. Many characterizations for modules with strongly -condition for strongly -type module are given. Also many connections between these types of module and some related types of modules are investigated.

The concept of epiform modules is a dual of the notion of monoform modules. In this work we give some properties of this class of modules. Also, we give conditions under which every hollow (copolyform) module is epiform.

The main goal of this paper is introducing and studying a new concept, which is named H-essential submodules, and we use it to construct another concept called Homessential modules. Several fundamental properties of these concepts are investigated, and other characterizations for each one of them is given. Moreover, many relationships of Homessential modules with other related concepts are studied such as Quasi-Dedekind, Uniform, Prime and Extending modules.

In this paper, we introduce and study a new concept named couniform modules, which is a dual notion of uniform modules, where an R-module M is said to be couniform if every proper submodule N of M is either zero or there exists a proper submodule N1 of N such that is small submodule of Also many relationships are given between this class of modules and other related classes of modules. Finally, we consider the hereditary property between R-module M and R-module R in case M is couniform.