On Morphic Properties Of Rings

The morphic rings were original from the study of unit regular rings, which havecancellation property for modules over them. Since morphic rings have many good prop-erties, such as internal cancellation property, and it has a very close relation with unitregular rings. In recent years, more and more algebraists devoted to the significant workof morphic rings.In the first chapter of this dissertation, some history facts and preliminary knowledgeabout morphic rings are introduced.The main context of chapter two is about the morphic property of polynomial ringof degree 1 with n indeterminates. Firstly, we study the relationship between multiplytrivial extension and polynomial ring of degree 1 with n indeterminates. The works aboutthe morphic properties of these polynomial rings are derived from that. Secondly, by thecharacterizing of unit elements [Claim 2.6] and a reduced representation of the elementsin that polynomial ring [Lemma 1], we give a fully characterization of morphic elementswhen R is strongly regular [Theorem 2.11].In chapter three, we give the new notion of centrally quasi-morphic ring, and studyits relation with centrally morphic ring which is firstly given by T.K. Li and Y. Zhou[11].We get that left centrally quasi-morphic rings are all directly finite [Proposition 3.11], thezero divisors of centrally quasi-morphic ring are commutative with each other [Proposi-tion 3.12], and centrally morphic ring is equivalent to centrally quasi-morphic ring [Theo-rem3.14]. At last, we research the relationship of centrally morphic rings, centrally quasi-morphic rings and regular rings. The main result is that a left centrally quasi-morphic ringR with J(R) = 0 is regular [Theorem 3.15]. Hence we get that centrally morphic rings,centrallyquasi-morphicringsandstronglyregularringsareequivalentincaseofJ(R) = 0[Proposition 3.17].