Semi-Regular Rings

Abstract

The study of relation between regularity and certain kinds of modules such as modules and modules، where ، is a modern subject in Rings and modules theory. Since every regular ring is semi-regular، we study  the relation between semi-regular ring and modules،  modules، where .

We prove that for every module ، the ring  is semi-regular if and only if is module and for every non-zero element ،  contains a non-zero direct summand of .

In addition، we proved that for every module ، the ring  is semi-regular if and only if  is module and for every non-zero element ، contained in a direct summand of .

Finally، we prove that if the ring  is semi-regular، then  is module and module.