Pure Maximal Submodules and Related Concepts

Abstract

      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-local) module an R-module  is named pure local (Pr-local) module. If it has only a Pr-maximal submodule which includes all proper submodule of . A ring  is named pure local (Pr-local) ring, if  is a Pr-local R-module. We give some relatio among Pr-maximal submodules and others related concept.