Research On Related Problems Of Morphic Rings

In this paper, we discuss two rings which are generalized morphicrings and pseudo-morphic rings. On one hand, we mainly discuss someproperties of generalized morphic rings and pseudo-morphic rings andthe relationship with other rings. On the other hand, we study trivialextension of generalized morphic rings and pseudo-morphic rings andthe necessary and sufcient condition of R[D, C] ring being generalizedmorphic rings or pseudo-morphic rings.In the first chapter, we mainly introduce some properties of gen-eralized morphic rings and the relationship with Kasch rings, quasi-morphic rings, P-injective rings. Moreover, we give new examples ofgeneralized morphic rings by trivial extension of a ring. We mainlyobtain some conclusions as follows:(1) Let R be a right generalizedmorphic ring. Then R is left quasi-morphic if and only if R is left gen-eralized morphic and right P-injective;(2) Suppose T=R∝V, whereR an integral domain and V a nonzero bimodule over R. Then T∝Tis not left generalized morphic.In the second chapter, we mainly discuss the relationship of pseudo-morphic rings and generalized morphic rings, the pseudo-morphic prop-erties of some elements in trivial extension. Also we give some examplesto explain that some inverse propositions need not true. Proving thefollowing conclusions:(1) If R is a pseudo-morphic ring, then R is ageneralized morphic ring. But generalized morphic ring may not be apseudo-morphic ring.(2) Let R be a ring, S=R∝R. If (0, a)∈S is a pseudo-morphic element, then a∈R is also pseudo-morphic. Con-versely, it is not true.In the last chapter, we mainly introduce the generalized morphicproperty and the pseudo-morphic property of the ring R[D, C]. Alsowe define left R[D, C] generalized morphic rings and left R[D, C] pseudo morphic rings. Proving the following conclusions: R[D, C] is aleft generalized morphic ring if and only if (1) D is a left generalizedmorphic ring;(2) for any x∈C, there exists y∈C such that lC(x)=Cy, lD(x)=Dy; R[D, C] is a left pseudo-morphic ring if and only if (1)D is a left pseudo-morphic ring;(2) for any x∈C, there exists y∈Csuch that C_x=lC(y), D_x=lD(y).