Nil-Cleanness Of Rings And Their Generalizations

A ring R is called(strongly)nil clean,if any element of R can be expressed as a sum of an idempotent and a nilpotent(that commute).Strongly nil clean rings are strongly clean and nil clean rings are clean.In this thesis,we study(uniquely)nil*-clean rings and strongly quasi-nil clean rings.And the paper consists of three parts.First,the background,research status,main work and basic concepts and symbols are presented in brief.Next,we extend*-rings to nil clean rings and define the(uniquely)nil*-clean rings.Basic properties and extended properties of(uniquely)nil*-clean rings are investigated.Also,the following results are proved:strongly nil*-clean rings are uniquely nil*-clean;uniquely nil*-clean rings are nil*-clean.Moreover,the relations among nil*-clean rings and related nil clean rings are discussed.Finally,we introduce strongly quasi-nil clean rings,which generalize the strongly nil clean rings.Some properties of strongly quasi-nil clean rings(elements)are given.And we prove that strongly nil clean elements are strongly quasi-nil clean and strongly quasi-nil clean elements are strongly clean.The structures of strongly quasi-nil clean rings are characterized.Furthermore,strong quasi-nil cleanness of generalized matrix rings over local rings is studied and under certain conditions,we discuss the relation between the strong quasi-nil cleanness and strong cleanness of generalized matrix rings.