Let R be commutative ring with identity and let M be any unitary left R-module. In this paper we study the properties of ec-closed submodules, ECS- modules and the relation between ECS-modules and other kinds of modules. Also, we study the direct sum of ECS-modules.

      In this paper, we introduce the notion of a 2-prime module as a generalization of prime module E over a ring R, where E is said to be prime module if (0) is a prime submodule. We introduced the concept of the 2-prime R-module. Module E is said to be 2-prime if (0) is 2-prime submodule of E. where a proper submodule K of module E is 2-prime submodule if, whenever rR, xE, E, Thus xK or [K: E].

        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 .

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.

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.

Let  be a module over a commutative ring  with identity. In this paper we intoduce the concept of Strongly Pseudo Nearly Semi-2-Absorbing submodule, where a  proper submodule  of an -module  is said to be Strongly Pseudo Nearly Semi-2-Absorbing submodule of   if whenever , for implies that either  or , this concept is a generalization of 2_Absorbing submodule, semi 2-Absorbing submodule, and strong form of (Nearly–2–Absorbing, Pseudo_2_Absorbing, and Nearly Semi–2–Absorbing) submodules. Several properties characterizations, and examples concerning this new notion are given. We study the relation between Strongly Pseudo Nearly Semei-2-Absorbing submodule and (2_Absorbing, Nearly_2_Absorbing, Pseudo_2_Absorbing, and Nearly S

Let R be associative ring with identity and M is a non- zero unitary left module over R. M is called M- hollow if every maximal submodule of M is small submodule of M. In this paper we study the properties of this kind of modules.

This paper is devoted to introduce weak and strong forms of fibrewise fuzzy ω-topological spaces, namely the fibrewise fuzzy -ω-topological spaces, weakly fibrewise fuzzy -ω-topological spaces and strongly fibrewise fuzzy -ω- topological spaces. Also, Several characterizations and properties of this class are also given as well. Finally, we focused on studying the relationship between weakly fibrewise fuzzy -ω-topological spaces and strongly fibrewise fuzzy -ω-topological spaces.

     The concept of a small f- subm was presented in a previous study. This work introduced a concept of a hollow f- module, where a module is said to be hollow fuzzy when every subm of it is a small f- subm. Some new types of hollow modules are provided namely, Loc- hollow f- modules as a strength of the hollow module, where every Loc- hollow f- module is a hollow module, but the converse is not true. Many properties and characterizations of these concepts are proved, also the relationship between all these types is researched. Many important results that explain this relationship are demonstrated also several characterizations and properties related to these concepts are given.

   Let  be a commutative ring with identity, and  be a unitary left R-module. In this paper we, introduce and study a new class of modules called pure hollow (Pr-hollow) and pure-lifting (Pr-lifting). We give a fundamental, properties of these concept.  also, we, introduce some conditions under which the quotient and direct sum of Pr-lifting modules is Pr-lifting.