In this work we discuss the concept of pure-maximal denoted by (Pr-maximal) submodules as a generalization to the type of R- maximal submodule, where a proper submodule  of an R-module  is called Pr- maximal if  ,for any submodule  of W is a pure submodule of W, We offer some properties of a Pr-maximal submodules, and we give Definition of the concept, near-maximal, a proper submodule  

 of an R-module  is named near (N-maximal) whensoever  is pure submodule of  such that  then K=.Al so we offer the concept Pr-module, An R-module W is named Pr-module, if every proper submodule of  is Pr-maximal. A ring  is named Pr-ring if whole proper ideal of  is a Pr-maximal ideal, we offer the concept pure local (Pr-loc

Let R be associative ring with identity and M is a non- zero unitary left module over R. M is called M- hollow if every maximal submodule of M is small submodule of M. In this paper we study the properties of this kind of modules.

The e-commerce is one of the best achievements of the twentieth century, since the conduct commercial transactions via the Internet may be the consumer easy selection process and purchase convenient manner different from traditional methods, and with the beginnings of the new millennium impose the emergence of e-commerce term significant challenges to the insurance industry as an important economic sectors Generally, and insurance companies in particular as a result of scientific development, which has led to a reduction in costs and innovation in the production, which led to intense competition on both levels local or global. The insurance industry is a vital part of the economy and it has a varied impact to the community and individual

Let R be an individual left R-module of the same type as W, with W being a ring containing one. W’s submodules N and K should be referred to as N and K, respectively that K ⊆ N ⊆ W if N/K <<_J (D_j (W)+K)/K, Then K is known as the D J-coessential submodule of Nin W as K⊆_ (Rce) N. Coessential submodule is a generalization of this idea. These submodules have certain interesting qualities, such that if a certain condition is met, the homomorphic image of D J- N has a coessential submodule called D J-coessential submodule.

  In this paper, we introduce and study the concept of S-coprime submodules, where a proper submodule N of an R-module M is called S-coprime submodule if M N is S-coprime Rmodule. Many properties about this concept are investigated.

Throughout this paper R represents a commutative ring with identity and all R-modules M are unitary left R-modules. In this work we introduce the notion of S-maximal submodules as a generalization of the class of maximal submodules, where a proper submodule N of an R-module M is called S-maximal, if whenever W is a semi essential submodule of M with N ? W ? M, implies that W = M. Various properties of an S-maximal submodule are considered, and we investigate some relationships between S-maximal submodules and some others related concepts such as almost maximal submodules and semimaximal submodules. Also, we study the behavior of S-maximal submodules in the class of multiplication modules. Farther more we give S-Jacobson radical of ri

Throughout this paper R represents a commutative ring with identity and all R-modules M are unitary left R-modules. In this work we introduce the notion of S-maximal submodules as a generalization of the class of maximal submodules, where a proper submodule N of an R-module M is called S-maximal, if whenever W is a semi essential submodule of M with N ⊊ W ⊆ M, implies that W = M. Various properties of an S-maximal submodule are considered, and we investigate some relationships between S-maximal submodules and some others related concepts such as almost maximal submodules and semimaximal submodules. Also, we study the behavior of S-maximal submodules in the class of multiplication modules. Farther more we give S-Jacobson radical of rings

      Let  be a right module over a ring  with identity. The semisecond submodules are studied in this paper. A nonzero submodule  of   is called semisecond if    for each . More information and characterizations about this concept is provided in our work.

     Let R be a commutative ring with unity and an R-submodule N is called semimaximal if and only if

 the sufficient conditions of F-submodules to be semimaximal .Also the concepts of (simple , semisimple) F- submodules and quotient F- modules are  introduced and given some  properties .