In this paper, we introduce and study the concept of a new class of generalized closed set which is called generalized b*-closed set in topological spaces ( briefly .g b*-closed) we study also. some of its  basic properties and investigate the relations between the associated topology. 

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

   In this paper, a new class of sets, namely - semi-regular closed sets is introduced and studied for topological spaces. This class properly contains the class of semi--closed sets and is property contained in the class of pre-semi-closed sets. Also, we introduce and study srcontinuity and sr-irresoleteness. We showed that sr-continuity falls strictly in between semi-- continuity and pre-semi-continuity.

The objective of this paper is to define and introduce a new type of nano semi-open set which called nano -open set as a strong form of nano semi-open set which is related to nano closed sets in nano topological spaces. In this paper, we find all forms of the family of nano -open sets in term of upper and lower approximations of sets and we can easily find nano -open sets and they are a gate to more study.  Several types of nano open sets are known, so we study relationship between the nano -open sets with the other known types of nano open sets in nano topological spaces. The Operators such as nano -interior and nano -closure are the part of this paper.

The significance fore supra topological spaces as a subject of study cannot be overstated, as they represent a broader framework than traditional topological spaces. Numerous scholars have proposed extension to supra open sets, including supra semi open sets, supra per open and others. In this research, a notion for ⱨ-supra open created within the generalizations of the supra topology of sets. Our investigation involves harnessing this style of sets to introduce modern notions in these spaces, specifically supra ⱨ - interior, supra ⱨ - closure, supra ⱨ - limit points, supra ⱨ - boundary points and supra ⱨ - exterior of sets. It has been examining the relationship with supra open. The research was also enriched with many

By use the notions pre-g-closedness and pre-g-openness we have generalized a class of separation axioms in topological spaces. In particular, we presented in this paper new types of regulαrities, which we named ρg­regulαrity and Sρg­regulαrity. Many results and properties of both types have been investigated and have illustrated by examples.

Let  be a module over a commutative ring  with identity. Before studying the concept of the Strongly Pseudo Nearly Semi-2-Absorbing submodule, we need to mention the ideal  and the basics that you need to study the  concept of the Strongly Pseudo Nearly Semi-2-Absorbing submodule. Also, we introduce several characteristics of the Strongly Pseudo Nearly Semi-2-Absorbing submodule in classes of multiplication modules and other types of modules. We also had no luck because the ideal  is not a Strongly Pseudo Nearly Semi-2-Absorbing ideal. Also, it is noted that  is the Strongly Pseudo Nearly Semi-2-Absorbing ideal under several conditions, which is this faithful module, projective module, Z-regular module and content module and non-si

Let  be a module over a commutative ring  with identity. In this paper we intoduce the concept of Strongly Pseudo Nearly Semi-2-Absorbing submodule, where a  proper submodule  of an -module  is said to be Strongly Pseudo Nearly Semi-2-Absorbing submodule of   if whenever , for implies that either  or , this concept is a generalization of 2_Absorbing submodule, semi 2-Absorbing submodule, and strong form of (Nearly–2–Absorbing, Pseudo_2_Absorbing, and Nearly Semi–2–Absorbing) submodules. Several properties characterizations, and examples concerning this new notion are given. We study the relation between Strongly Pseudo Nearly Semei-2-Absorbing submodule and (2_Absorbing, Nearly_2_Absorbing, Pseudo_2_Absorbing, and Nearly S

     The objective of this paper is to show modern class of open sets which is an -open. Some functions via this concept were studied and the relationships such as continuous function strongly -continuous function -irresolute function -continuous function.

E-learning seeks to create an interactive learning environment between the teacher and the learner through electronic media conveying in more than one direction, regardless of how the environment and its variables are identified. It also develops skills necessary to deal with technology in order to be able to take into account the individual differences between them and helps e-learning teacher and learner to achieve the goals set in advance and identify educational objectives in a clear manner. The research aims to identify e-learning in its benefits and management systems. It has three sections dealt with in the current research. Chapter II concentrates on the research Methodology, which consisted of three sections: The first s