Let  be an R-module, and let  be a submodule of . A submodule  is called -Small submodule () if for every submodule  of  such that  implies that . In our work we give the definition of -coclosed submodule and -hollow-lifiting modules with many properties.

        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.

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.

     In this paper we introduce the notions of t-stable extending and strongly t-stable extending modules. We investigate properties and characterizations of each of these concepts. It is shown that a direct sum of t-stable extending modules is t-stable extending while with certain conditions a direct sum of strongly t-stable extending is strongly t-stable extending. Also, it is proved that under certain condition, a stable submodule of t-stable extending (strongly t-stable extending) inherits the property.

      Let S be a commutative ring with identity, and A is an S-module. This paper introduced an important concept, namely strongly maximal submodule. Some properties and many results were proved as well as the behavior of that concept with its localization was studied and shown.

      In this work we discuss the concept of pure-maximal denoted by (Pr-maximal) submodules as a generalization to the type of R- maximal submodule, where a proper submodule  of an R-module  is called Pr- maximal if  ,for any submodule  of W is a pure submodule of W, We offer some properties of a Pr-maximal submodules, and we give Definition of the concept, near-maximal, a proper submodule  

 of an R-module  is named near (N-maximal) whensoever  is pure submodule of  such that  then K=.Al so we offer the concept Pr-module, An R-module W is named Pr-module, if every proper submodule of  is Pr-maximal. A ring  is named Pr-ring if whole proper ideal of  is a Pr-maximal ideal, we offer the concept pure local (Pr-loc

A standard theoretical neutron energy flux distribution is achieved for the triton-triton nuclear fusion reaction in the range of triton energy about ≤10 MeV. This distribution give raises an evidence to provide the global calculations including the characteristics fusion parameters governing the T-T fusion reaction.

Let A ⊆ V(H) of any graph H, every node w of H be labeled using a set of numbers; , where d(w,v) denotes the distance between node w and the node v in H, known as its open A-distance pattern. A graph H is known as the open distance-pattern uniform (odpu)-graph, if there is a nonempty subset A ⊆V(H) together with  is the same for all . Here  is known as the open distance pattern uniform (odpu-) labeling of the graph H and A is known as an odpu-set of H. The minimum cardinality of vertices in any odpu-set of H, if it exists, will be known as the odpu-number of the graph H. This article gives a characterization of maximal outerplanar-odpu graphs. Also, it establishes that the possible odpu-number of an odpu-maximal outerplanar graph i

        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.

This paper generalizes and improves the results of Margenstren, by proving that the number of -practical numbers  which is defined by   has a lower bound in terms of . This bound is more sharper than Mangenstern bound when  Further general results are given for the existence of -practical numbers, by proving that the interval contains a -practical for all