Group actions in a regular ring

Abstract

Let R be a ring with identity, X the set of all nonzero,nonunits of R and G the group of all units of R. We will considertwo group actions on X by G, the regular action and the conjugateaction. In this paper, by investigating two group actions we canhave some results as follows: First, if G is a nitely generatedabelian group, then the orbit O(x) under the regular action on X by G is nite for all nilpotents x 2 X . Secondly, if F is a eld in which 2 is a unit and F-{0} is a finitley generated abelian group,then F is nite. Finally, ifG in a unit-regular ring R is a torsiongroup and 2 is a unit in R, then the conjugate action on X by G is trivial if and only if G is abelian if and only if R is commutative.