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.

        Throughout this paper, three concepts are introduced namely stable semisimple modules, stable t-semisimple modules and strongly stable t-semisimple. Many features co-related with these concepts are presented. Also many connections between these concepts are given. Moreover several relationships between these classes of modules and other co-related classes and other related concepts are introduced.

     The -s-extending modules will be purpose of this paper, a module M  is -s-extending if each submodule in M is essential in submodule has a supplement that is direct summand. Initially, we give relation between this concept with weakly supplement extending modules and -supplemented modules. In fact, we gives the following implications:

Lifting modules   -supplemented modules   -s-extending modules  weakly supplement extending modules.

It is also we give examples show that, the converse of this result is not true. Moreover, we study when the converse of this result is true.

We introduce the notion of t-polyform modules. The class of t- polyform modules contains the class of polyform modules and contains the class of t-essential quasi-Dedekind.

     Many characterizations of t-polyform modules are given. Also many connections between these class of modules and other types of modules are introduced.

Let
be an
module, and let
be a set, let
be a soft set over
. Then
is said to be a fuzzy soft module over
iff
,
is a fuzzy submodule of
. In this paper, we introduce the concept of fuzzy soft modules over fuzzy soft rings and some of its properties and we define the concepts of quotient module, product and coproduct operations in the category of
modules.

The aim of this paper is introducing the concept of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal. Some properties of (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal have been studied and another characterizations have been given. The relationship of (ɱ,ɳ) strong full stability B-Algebra-module related to an ideal that states,  a B- -module Ӽ is (ɱ,ɳ)- strong full stability B-Algebra-module related to an ideal  , if and only if  for any two ɱ-element sub-sets and of Ӽ^ɳ, if , for each j = 1, …, ɱ,  i = 1,…, ɳ  and   implies _Ạɳ( ) _Ạɳ(  have been proved..

An R-module M is called rationally extending if each submodule of M is rational in a direct summand of M. In this paper we study this class of modules which is contained in the class of extending modules, Also we consider the class of strongly quasi-monoform modules, an R-module M is called strongly quasi-monoform if every nonzero proper submodule of M is quasi-invertible relative to some direct summand of M. Conditions are investigated to identify between these classes. Several properties are considered for such modules

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.

Let R be a commutative ring with identity, and let M be a unitary (left) R- modul e. The ideal annRM  = {r E R;rm  = 0 V  mE M} plays a central

role  in  our  work.  In  fact,  we  shall  be  concemed   with  the  case  where annR1i1 = annR(x) for   some   x EM such  modules  will  be  called bounded  modules.[t  htrns out that there are many classes of modules properly contained in the class of bounded modules such as cyclic modules, torsion -G·ee modulcs,faithful  multiplicat