pi-regular rings

All rings studied in this dissertation are associative with 1 {dollar}ne{dollar} 0. A ring R is called semicommutative if for every a,b{dollar}epsilon{dollar}R, there exist h,f{dollar}epsilon{dollar}R such that ab = ha and ba = af. A ring R is called regular if for every x{dollar}epsilon{dollar}R there exists y{dollar}epsilon{dollar}R such that xyx=x. Where R is called {dollar}pi{dollar}-regular if for every x{dollar}epsilon{dollar}R there exists n {dollar}ge{dollar} 1 such that x{dollar}sp{lcub}rm n{rcub}{dollar} is regular.; The dissertation focuses on the structure of {dollar}pi{dollar}-regular (regular) rings. In chapter two we show that a semicommutative ring R is {dollar}pi{dollar}-regular if and only if R/Nil(R) is regular, where Nil(R) is the two-sided ideal of all nilpotent elements in R. In chapter three we give a necessary and sufficient condition on a ring R so that it is {dollar}pi{dollar}-regular. In chapter four we state a necessary and sufficient condition on a {dollar}pi{dollar}-regular ring R so that it is semicommutative. In chapter five we show that R-{dollar}{lcub}0{rcub}{dollar} is a union of disjoint multiplicative groups if and only if R is semicommutative regular. In chapter six under the assumption that 2 is a unit in a ring R, we discover that the idempotents of R are in the center of R if and only if the set of units {dollar}{lcub}{dollar}x{dollar}epsilon{dollar}R: x{dollar}sp2{dollar} = 1{dollar}{rcub}{dollar} is a multiplicative subgroup in R and consequently we show that a regular ring R is commutative if and only if the set of units of R is commutative. In the last chapter we give a short proof of a well-known result on matrices and we state its relation with {dollar}pi{dollar}-regular rings.