We use the idea of the grill. This study generalized a new sort of linked space like -connected and -hyperconnected and investigated its features, as well as the relationship between it and previously described notions. It also developed new sorts of functions, such as hyperconnected space, and identified their relationship by offering numerous instances and attributes that belong to this set. This set will serve as a starting point for further research into the set many future possibilities. We also use some theorems and observations previously studied and related to the grill and the semi-open to obtain results in this research. We applied the concept of connected to them and obtained results related to connected. The sources related t

In this paper we introduced a new class of - called - and study their basic properties in nano topological spaces. We also introduce -closure and -interior and study some of their fundamental properties.

New types of modules named Fully Small Dual Stable Modules and Principally Small Dual Stable are studied and investigated. Both concepts are generalizations of Fully Dual Stable Modules and Principally Dual Stable Modules respectively. Our new concepts coincide when the module is Small Quasi-Projective, and by considering other kind of conditions. Characterizations and relations of these concepts and the concept of Small Duo Modules are investigated, where every fully small dual stable R-module M is small duo and the same for principally small dual stable.

      Let R be a commutative ring with unity. And let E be a unitary R-module. This paper introduces the notion of 2-prime submodules as a generalized concept of 2-prime ideal, where proper submodule H of module F over a ring R is said to be 2-prime if , for r R and x F implies that  or . we prove many properties for this kind of submodules, Let H is a submodule of module F over a ring R then H is a 2-prime submodule if and only if [N ] is a 2-prime submodule of E, where r R. Also, we prove that if F is a non-zero multiplication module, then [K: F] [H: F] for every submodule k of F such that H K. Furthermore, we will study the basic properties of this kind of submodules.

Let R be a ring and let M be a left R-module. In this paper introduce a small pointwise M-projective module as generalization of small M- projective module, also introduce the notation of small pointwise projective cover and study their basic properties.
.