A STUDY ON GENERALIZATIONS OF UN AND VNL RINGS AND GROUP RINGS

Abstract

The main aim of this thesis is to study generalizations of UN and VNL rings, as well as group rings. As an introduction to this thesis, the introductory chapter collects literature and definitions relevant to each concept that are used throughout this thesis. Calug ˘ areanu in [7] introduced and investigated UN rings. A ring ˘ R is called UN ring if every non unit of it can be written as product of a unit and a nilpotent element. We carry his study of UN rings further. We have focused on UN group rings. In our study of UN group rings, necessary and sufficient conditions for group ring RG to be UN have been obtained. We have studied a generalization of the class of UN rings, called UQ rings. A ring R is called UQ ring if every non unit element of R can be represented as a product of a unit and a quasiregular element. Various properties of these rings along with its characterizations are obtained and examples are provided to show that the class of UQ rings properly contains classes of UN rings, J-UN rings and 2-good rings. In study of UQ group rings, necessary and sufficient conditions for commutative group ring RG to be UQ have been established. An element a of R is called SWR if a ∈ aRa2R. A ring R is called an almost SWR if for any a ∈ R, either a or 1 − a is SWR. The class of almost SWR rings properly contains the classes of SWR and abelian VNL rings. Various properties of almost SWR are obtained. We provide characterizations of almost SWR rings. Further, we study SWR group rings and almost SWR group rings. If a ring, R, satisfies the condition that its every proper homomorphic image has a certain property P, then the ring R is called restricted P ring. This has motivated us to introduce and investigate a new class of rings called semiboolean neat rings. The ring R is semiboolean neat provided that every proper homomorphic image of R is semiboolean. The class of semiboolean neat rings lies strictly between the classes of nil neat and neat rings. We obtain characterizations of semiboolean neat rings. Moreover, commutative semiboolean neat group rings have also been studied.A ring R is said to be weakly g(x)-invo clean if each element of R is either a sum or difference of an involution and a root of g(x). This class is a proper subclass of weakly g(x)-clean rings and a generalization of g(x)-invo clean rings. Various proper ties of weakly g(x)-invo clean rings are given. We determine necessary and sufficient conditions for skew Hurwitz series ring (HR, α) to be weakly g(x)-invo clean, where α is an endomorphism of R. Finally, the last chapter summarizes the thesis with a brief conclusion and discusses some future prospects.