The Regular Rings With Strongly Injective

Since the last century,regularity of rings has become an important topic in ring theory.The research methods of regularity of rings can be divided into two kinds:One is to understand rings by studying the inner structure and the properties of rings;The other is to use module theory to characterize the outer part of a ring by studying the properties of modules on the ring.In this paper,we study regular rings using strongly complete principal injective modes.This paper is composed of four parts.In the first part,the research background of this paper is introduced,the research status of regular rings and completely principally injective modules at home and abroad is briefly described,and the main research content and significance of this paper are summarized.In the second part,the basic concepts and lemmas used in this paper are introduced.In the third part,the properties of regular rings are generalized,and the equivalence conditions of regular rings and weakly regular rings are studied.In the fourth part,the equivalence conditions of regular rings,weakly regular rings,strongly bounded rings and strongly left completely principally injective rings are studied by using the equivalence conditions of regular rings and weakly regular rings.This section consists of three parts.First,the equivalence conditions of left completely principally injective rings and strongly left completely principally injective rings are explored.Secondly,the equivalence conditions of regular rings and strongly left completely principally injective rings are studied.Finally,by using the equivalence conditions of regular rings and strongly regular rings,the equivalence conditions of strongly regular rings and strongly left completely principally injective rings are studied.By studying the equivalence conditions of regular rings and strongly bounded rings,strongly left completely principally injective rings,the equivalence conditions of regularity,weak regularity and strongly bounded,and strongly left completely principally injective rings are obtained.