A new generalizations of coretractable modules are introduced where a module is called t-essentially (weakly t-essentially) coretractable if for all proper submodule of , there exists f End( ), f( )=0 and Imf tes (Im f + tes ). Some basic properties are studied and many relationships between these classes and other related one are presented.
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.
Let R be a ring with identity and M be a right unitary R-module. In this paper we
introduce the notion of strongly coretractable modules. Some basic properties of this
class of modules are investigated and some relationships between these modules and
other related concepts are introduced.
Let R be a commutative ring with 1 and M be a left unitary R-module. In this paper, we give a generalization for the notions of compressible (retractable) Modules. We study s-essentially compressible (s-essentially retractable). We give some of their advantages, properties, characterizations and examples. We also study the relation between s-essentially compressible (s-essentially retractable modules) and some classes of modules.
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 a commutative ring with 1 and be a left unitary . In this paper, the generalizations for the notions of compressible module and retractable module are given.
An is termed to be semi-essentially compressible if can be embedded in every of a non-zero semi-essential submodules. An is termed a semi-essentially retractable module, if for every non-zero semi-essentially submodule of an . Some of their advantages characterizations and examples are given. We also study the relation between these classes and some other classes of modules.
In this paper, as generalization of second modules we introduce type of modules namely (essentially second modules). A comprehensive study of this class of modules is given, also many results concerned with this type and other related modules presented.
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 .
Let R be a commutative ring with 1 and M be a (left) unitary R – module. This essay gives generalizations for the notions prime module and some concepts related to it. We termed an R – module M as semi-essentially prime if annR (M) = annR (N) for every non-zero semi-essential submodules N of M. Given some of their advantages characterizations and examples, and we study the relation between these and some classes of modules.
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.
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.