Bull. Korean Math. Soc. 2005 Vol. 42, No. 4, 807-815 Published online December 1, 2005

Juncheol Han Pusan National University

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 consider two group actions on $X$ by $G$, the regular action and the conjugate action. In this paper, by investigating two group actions we can have some results as follows: First, if $G$ is a finitely generated abelian group, then the orbit $O(x)$ under the regular action on $X$ by $G$ is finite for all nilpotents $x \in X$. Secondly, if $F$ is a field in which 2 is a unit and $F \setminus \{0\}$ is a finitley generated abelian group, then $F$ is finite. Finally, if $G$ in a unit-regular ring $R$ is a torsion group 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.