Let  be a connected graph with vertices set  and edges set . The ordinary distance between any two vertices of  is a mapping  from  into a nonnegative integer number such that  is the length of a shortest  path. The maximum distance between two subsets  and  of   is the maximum distance between any two vertices  and  such that  belong to  and  belong to . In this paper, we take a special case of maximum distance when  consists of one vertex and  consists of  vertices, . This distance is defined by: where  is the order of  a graph .

     In this paper, we defined  – polynomials based on

     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.

A non-zero module M is called hollow, if every proper submodule of M is small. In this work we introduce a generalization of this type of modules; we call it prime hollow modules. Some main properties of this kind of modules are investigated and the relation between these modules with hollow modules and some other modules are studied, such as semihollow, amply supplemented and lifting modules.

        Let R be a commutative ring with unity. In this paper we introduce the notion of chained fuzzy modules as a generalization of chained modules. We investigate several characterizations and properties of this concept

      In this paper, we introduce the notion of a 2-prime module as a generalization of prime module E over a ring R, where E is said to be prime module if (0) is a prime submodule. We introduced the concept of the 2-prime R-module. Module E is said to be 2-prime if (0) is 2-prime submodule of E. where a proper submodule K of module E is 2-prime submodule if, whenever rR, xE, E, Thus xK or [K: E].

     This paper aims to introduce the concepts of  -closed, -coclosed, and -extending modules as generalizations of the closed, coclossed, and extending modules,  respectively. We will prove some properties as when the image of the e*-closed submodule is also e*-closed and when the submodule of the e*-extending module is e*-extending. Under isomorphism, the e*-extending modules are closed. We will study the quotient of e*-closed and e*-extending, the direct sum of e*-closed, and the direct sum of e*-extending.

Throughout this paper we introduce the concept of quasi closed submodules which is weaker than the concept of closed submodules. By using this concept we define the class of fully extending modules, where an R-module M is called fully extending if every quasi closed submodule of M is a direct summand.This class of modules is stronger than the class of extending modules. Many results about this concept are given, also many relationships with other related concepts are introduced.

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.