The goal of this discussion is to study the twigged of pure-small (pr-small) sub- moduleof a module W as recirculation of a small sub-module, and we give some basic idiosyncrasy and instances of this kind of sub-module. Also, we give the acquaint of pure radical of a module W (pr-radical) with peculiarities.

        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, we introduce the concept of s.p-semisimple module. Let S be a semiradical property, we say that a module M is s.p - semisimple if for every submodule N of M, there exists a direct summand K of M such that K ≤ N and N / K has S. we prove that a module M is s.p - semisimple module if and only if for every submodule A of M, there exists a direct summand B of M such that A = B + C and C has S. Also, we prove that for a module M is s.p - semisimple if and only if for every submodule A of M, there exists an idempotent e ∊ End(M) such that e(M) ≤ A and (1- e)(A) has S. 

Let R be a Γ-ring and G be an R_Γ-module. A proper R_Γ-submodule S of G is said to be semiprime R_Γ-submodule if for any ideal I of a Γ-ring R and for any R_Γ-submodule A of G such that or which implies that . The purpose of this paper is to introduce interesting results of semiprime R_Γ-submodule of R_Γ-module which represents a generalization of semiprime submodules.

        Throughout this paper, we introduce the notion of weak essential F-submodules of F-modules as a generalization of  weak essential submodules. Also we study the homomorphic image and inverse image of weak essential F-submodules.

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.

 Abstract

In order to determine what type of photovoltaic solar module could best be used in a thermoelectric photovoltaic power generation. Changing in powers due to higher temperatures (25^oC, 35^oC, and 45^oC) have been done for three types of solar modules: monocrystalline , polycrystalline, and copper indium gallium (di) selenide (CIGS). The Prova 200 solar panel analyzer is used for the professional testing of three solar modules at different ambient temperatures; 25^oC, 35^oC, and 45^oC and solar radiation range 100-1000 W/m². Copper indium gallium (di) selenide module   has the lowest power drop (with the average percent

The concept of St-Polyform modules, was introduced and studied by Ahmed in [1], where a module M is called St-polyform, if for every submodule N of M and for any homomorphism 𝑓:N M; ker𝑓 is St-closed submodule in N. The novelty of this paper is to dualize this class of modules, the authors call it CSt-polyform modules, and according to this dualizations, some results which appeared in [1] are dualized for example we prove that in the class of hollow modules, every CSt-polyform module is coquasi-Dedekind. In addition, several important properties of CSt-polyform module are established, and other characterization of CSt-polyform is given. Moreover, many relationships of CSt-polyform modules with other related concepts are

The concepts of generalized higher derivations, Jordan generalized higher derivations, and Jordan generalized triple higher derivations on Γ-ring M into ΓM-modules X are presented. We prove that every Jordan generalized higher derivation of Γ-ring M into 2-torsion free ΓM-module X, such that aαbβc=aβbαc, for all a, b, c M and α,βΓ, is Jordan generalized triple higher derivation of M into X.

In this paper, we present the almost approximately nearly quasi compactly packed (submodules) modules as an application of the almost approximately nearly quasiprime submodule. We give some examples, remarks, and properties of this concept. Also, as the strong form of this concept, we introduce the strongly, almost approximately nearly quasi compactly packed (submodules) modules. Moreover, we present the definitions of almost approximately nearly quasiprime radical submodules and almost approximately nearly quasiprime radical submodules and give some basic properties of these concepts that will be needed in section four of this research. We study these two concepts extensively.