This paper intends to initiate a new type of generalized closed set in topological space with the theoretical application of generalized topological space. This newly defined set is a weaker form than the -closed set as well as -closed set. Some phenomenal characterizations and results of newly defined sets are inculcated in a proper manner. The characteristics of normal spaces and regular spaces are achieved in the light of the generalized pre-regular closed set.

There are two (non-equivalent) generalizations of Von Neuman regular rings to modules; one in the sense of Zelmanowize which is elementwise generalization, and the other in the sense of Fieldhowse. In this work, we introduced and studied the approximately regular modules, as well as many properties and characterizations are considered, also we study the relation between them by using approximately pointwise-projective modules.

  In this paper we introduced the concept of 2-pure submodules as a generalization of pure submodules, we study some of its basic properties and by using this concept we define the class of 2-regular modules, where an R-module M is called 2-regular module if every submodule is 2-pure submodule. Many results about this concept are given. 

An R-module M is called a 2-regular module if every submodule N of M is 2-pure submodule, where a submodule N of M is 2-pure in M if for every ideal I of R, I2MN = I2N, [1]. This paper is a continuation of [1]. We give some conditions to characterize this class of modules, also many relationships with other related concepts are introduced.

We introduce in this paper the concept of an approximately pure submodule as a     generalization of a pure submodule, that is defined by Anderson and Fuller. If every submodule of an R-module  is approximately pure, then  is called F-approximately regular. Further, many results about this concept are given.

           This paper introduce two types of edge degrees (line degree and near line degree) and total edge degrees (total line degree and total near line degree) of an edge in a fuzzy semigraph, where a fuzzy semigraph is defined as (V, σ, μ, η) defined on a semigraph G^* in which σ : V → [0, 1], μ : VxV → [0, 1] and η : X → [0, 1] satisfy the conditions that for all the vertices u, v in the vertex set,  μ(u, v) ≤ σ(u) ᴧ σ(v) and  η(e) = μ(u₁, u₂) ᴧ μ(u₂, u₃) ᴧ … ᴧ μ(u_n-1, u_n) ≤ σ(u₁) ᴧ σ(u_n), if e = (u₁, u₂, …, u_n), n ≥ 2 is an edge in the semigraph G

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

    This paper devoted to the analysis of regular singular initial value problems for ordinary differential equations with a singularity of the first kind , we propose semi - analytic technique using two point osculatory interpolation to construct polynomial solution, and discussion behavior of the solution in the neighborhood of the regular singular points and its numerical approximation, two examples are presented to demonstrate the applicability and efficiency of the methods. Finally , we discuss behavior of the solution in the neighborhood of the singularity point which appears to perform satisfactorily for singular problems.

   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.

A submoduleA of amodule M is said to be strongly pure , if for each finite subset {ai} in A , (equivalently, for each a ?A) there exists ahomomorphism f : M ?A such that f(ai) = ai, ?i(f(a)=a).A module M is said to be strongly F–regular if each submodule of M is strongly pure .The main purpose of this paper is to develop the properties of strongly F–regular modules and study modules with the property that the intersection of any two strongly pure submodules is strongly pure .