Generalizations Of Duo Rings And Their Applications

In this thesis, we investigate2-QD rings and weakly-abelian rings, both of which are generalizations of duo rings, and they possess various properties and extensive applications. Many results on duo and quasi-duo rings will be extended to such rings. This thesis is divided into four parts:Firstly, we make a introduction to duo rings as well as related ones developed from it. Further, a survey of this thesis is given here.Second, a series of preliminary definitions and results are listed so as to read this article efficiently.Third, we introduce2-QD ring and show that duo rings, quasi-duo rings and NQD rings are2-QD while the converse does not hold by counter-examples. Sev-eral properties and theorems of duo rings are generalized to2-QD rings and many new characterizations are given. Furthermore, we discuss the relations among2-QD rings, reduced rings, reversible rings, semi-commutative rings as well as symmetric rings. It is shown that the previous classes of rings coincide with one another under semiprimitive case.Then, we discuss abelian rings, and derive many new characteristics of such rings. Further, we concentrate on weakly-abelian ring. We will consider hereditary properties of such rings to many extensions such as trival extension, polynomial extension, power series extension, upper-triangular matrix extension, Dorroh ex-tension and Nagata extension. Rings with endomorphisms under weakly-abelian conditions are also investigated. The relations among weakly-abelian rings and many others are studied as well.After that, we listed several problems which should be considered in the future.