Group actions in a unit-regular ring with commuting idempotents

Abstract

Let R be a ring with unity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will consider some group actions on X by G, the left (resp. right) regular action and the conjugate action. In this paper, by investigating these group actions we can have some results as follows: First, if E(R), the set of all nonzero nonunit idempotents of a unit-regular ring R, is commuting, then ol(x) = or(x), oc(x) = {x} for all x ∈ X where ol(x) (resp. or(x), oc(x)) is the orbit of x under the left regular (resp. right regular, conjugate) action on X by G and R is abelian regular. Secondly, if R is a unit-regular ring with unity 1 such that G is a cyclic group and 2 = 1 + 1 ∈ G, then G is a finite group. Finally, if R is an abelian regular ring such that G is an abelian group, then R is a commutative ring.