On Strongly ?-Regular Group Rings

Abstract

Let R be an associative ring with unit. An element x ∈ R is said to be left (right) π-regular if there exist y ∈ R and a positive integer n such that $$x^{n}=yx^{n+1}(x^{n}=x^{n+1}y)$$ . If x is both left and right π-regular, then it is said to be strongly π-regular. R is said to be a strongly π-regular ring if all its elements are strongly π-regular. In this paper we determine some conditions which are necessary or sufficient for a group ring to be strongly π-regular.