Some Questions About Quasi-morphic Ring And Quasi-morphic Group

In 1976,in the paper [1], Erlich showed that an endomorphismαof a module M isunit regular if and only if it is regular and M/im(α) ker(α). The notion of morphic ringsgrow out of this characterization. The ring R is a left morphic, if every element a in RsatisfiesR/Ra l(a). Inrecentyears, mrophicringsattractedmoreandmoreattentionsofexperts of ring theory. And may new notions, such as quasi-morphic,π-morphic and G-morphicweredevelopedfromthat. In2005,NicholsonandCamposextendedthemorphicpropertyofringstomodules,theyobtainedsomesignificantresults. Themorphicpropertycan also be extended to groups naturally, by regarding groups as Z-modules.The present dissertation focus on the morphic property of rings, and the relationshipamong morphic, quasi-morphic and regular property. In the last chapter, a new notionquasi-morphic group will be defined. And some of our works about it are included in thesame chapter.In chapter two, based on a result of us about unit regular, we give a sufficient andnecessary condition for a ring to be strongly regular. In this chapter, we also study theproperty of reduced morphic rings, we prove that for reduced ring, strongly regular, unitregular, regular, quasi-morphic and morphic are equivalent.We study quasi-morphic rings in chapter 3. One of the main result of the paper [9]is that finite intersections and finite sums of principal left ideals of left quasi-morphicrings are again principal. In this chapter, firstly by an counter-example, we show thatinfinite sums of principal ideals of quasi-morphic rings is not always principal ideal. It'swell know that, in general case, quasi-morphic property is not Morita invariant, thoughwhen R is a regular or a semi-simple ring, the matrix rings of R are quasi-morphic. In thenext party of chapter 3, by the study of quasi-morphic modules, we give a sufficient andnecessary condition for the matrix rings of a quasi-morphic ring R to be quasi-morphic.We also prove that the formal triangular matrix ring T of an (A, B)-bimodule M is quasi-morphic if and only if A, B are quasi-morphic rings and M = 0. Based on the result, someproperty of corner rings of a quasi-morphic is obtained.In the last chapter, we give a new notion named quasi-morphic group, and give someequivalent characterization of that group in the same time. We study the quasi-morphicproperty of some special groups in this chapter. We get that, when an abelian group is quasi-morphic, the sets of it's images and the sets of it's kernels are the same, which is agrape. In the last part of this chapter, we give a sufficient and necessary condition for afinitely generated abelian group to be quasi-morphic.