Let R be a commutative ring with identity, and let M be a unitary left R-module. M is called Z-regular if every cyclic submodule (equivalently every finitely generated) is projective and direct summand. And a module M is F-regular if every submodule of M is pure. In this paper we study a class of modules lies between Z-regular and F-regular module, we call these modules regular modules.

Abstract In this work we introduce the concept of approximately regular ring as generalizations of regular ring, and the sense of a Z- approximately regular module as generalizations of Z- regular module. We give many result about this concept.

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 .

           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

It is shown that if a subset of a topological space (χ, τ) is δ-semi.closed, then it is semi.closed. By use this fact, we introduce the concept regularity of a topological space (χ, τ) via δ-semi.open sets. Many properties and results were investigated and studied. In addition we study some maps that preserve the δ-semi.regularity of spaces.

In this paper, certain types of regularity of topological spaces have been highlighted, which fall within the study of generalizations of separation axioms. One of the important axioms of separation is what is called regularity, and the spaces that have this property are not few, and the most important of these spaces are Euclidean spaces. Therefore, limiting this important concept to topology is within a narrow framework, which necessitates the use of generalized open sets to obtain more good characteristics and preserve the properties achieved in general topology. Perhaps the reader will realize through the research that our generalization preserved most of the characteristics, the most important of which is the hereditary property. Two t

        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 commutative ring with unity 1 6= 0, and let M be a unitary left module over R. In this paper we introduce the notion of epiform∗ modules. Various properties of this class of modules are given and some relationships between these modules and other related modules are introduced.