Let be a commutative ring with unity and let be a non-zero unitary module. In
this work we present a -small projective module concept as a generalization of small
projective. Also we generalize some properties of small epimorphism to δ-small
epimorphism. We also introduce the notation of δ-small hereditary modules and δ-small
projective covers.

      In this paper, we extend the work of our proplem in uniformly convex Banach spaces using Kirk fixed point theorem. Thus the existence and sufficient conditions for the controllability to general formulation of nonlinear boundary control problems in reflexive Banach spaces are introduced. The results are obtained by using fixed point theorem that deals with nonexpanisive mapping defined on a set has normal structure and strongly continuous semigroup theory. An application is given to illustrate the  importance of the results.

     This paper aims to introduce the concepts of  -closed, -coclosed, and -extending modules as generalizations of the closed, coclossed, and extending modules,  respectively. We will prove some properties as when the image of the e*-closed submodule is also e*-closed and when the submodule of the e*-extending module is e*-extending. Under isomorphism, the e*-extending modules are closed. We will study the quotient of e*-closed and e*-extending, the direct sum of e*-closed, and the direct sum of e*-extending.

Let  be a commutative ring with an identity and be a unitary -module. We say that a non-zero submodule  of  is  primary if for each with en either or  and an -module  is a small primary if   =  for each proper submodule  small in. We provided and demonstrated some of the characterizations and features of these types of submodules (modules).  

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 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 .
 

In this paper, we introduce the concept of e-small M-Projective modules as a generalization of M-Projective modules.

The aim of this paper is to study the Zariski topology of a commutative KU-algebra. Firstly, we introduce new concepts of a KU-algebra, such as KU-lattice, involutory ideal and prime ideal and investigate some basic properties of these concepts. Secondly, the notion of the topology spectrum of a commutative KU-algebra is studied and several properties of this topology are provided. Also, we study the continuous map of this topological space.

Throughout this note, R is commutative ring with identity and M is a unitary R-module. In this paper, we introduce the concept of quasi J-  submodules as a     –  and give some of its basic properties. Using this concept, we define the class of quasi J-regular modules, where an R-module     J- module if every submodule of  is quasi J-pure. Many results about this concept

In this research, we introduce a small essentially quasi−Dedekind R-module to generalize the term of an essentially quasi.−Dedekind R-module. We also give some of the basic properties and a number of examples that illustrate these properties.