Morphic Properties And Its Generalizations

In this thesis, we mainly investigate the structure of morphic rings and their extensions. That is, we will develop several new structures for related rings and modules.We have seven parts in this article.The first part:We introduce some known results on morphic rings and modules and give a survey of my work.The second part:We give some preliminaries which will be used in the sequel.The third part:Firstly, we introduce a class of rings between morphic rings andπ-morphic rings, that is, n-morphic rings. We investigate the relations between n-morphic rings and other special rings. It is shown that the corner ring of an n-morphic ring is also an n-morphic ring, and provided some counter examples to show an n-morphic ring not being m-morphic (mn, m,n∈N,n≥2) and aπ-morphic ring not being n-morphic. Secondly, we obtain the equivalent characterization of n-morphic modules. The relations between n-morphic modules and their endomorphism rings are studied.The fourth part:We define the subring-extensions of rings, especially discuss the morphic properties of subring-extensions.The fifth part:Two specific rings R[D] andτ[D] are constructed, which is said to be a matrix or diagonal tail ring over D, respectively. The morphic properties of the two different classes of rings coincide with each other.The sixth part:We introduce and investigate ML-rings and ML-modules. The main results are:(1) Let Ri(i∈I) be associative rings with identity. Then (?)Ri is a left ML-ring if and only if there exists i0∈I such that Ri0 is a left ML-ring and for each ,i∈I-{i0}, Ri is a left morphic ring. (2) Let R be associative rings with identity. If R∝R is a left ML-ring, then so is R. But the converse is not true.(3) If n≥2 and n=p1r1p2r2…psrs is a prime power decomposition of n, thenΖn∝Ζn is a left ML-ring if and only if ri>1 for at most one value of i if and only ifΖn is a VNL-ring.The seven part:We study the morphic properties of two special classes of 3×3 matrices in the forms and including some special elements in them. We show the following assertions.(1) Neither (?)(R) nor (?)(R) is left morphic.(2) Let R be a uniquely morphic ring and R∝R be a left morphic ring. Then the elements whose principal diagonal are nonzero in R are left morphic in (?)(R).