CONJUGATE ACTION IN A LEFT ARTINIAN RING

IF R is a left Artinian ring with identity, G is the group of units of R and X is the set of nonzero, nonunits of R, then G acts naturally on X by conjugation. It is shown that if the conjugate action on X by G is trivial, that is, gx = xg for all $g \in G$ and all $x \in X$, then R is a commutative ring. It is also shown that if the conjegate action on X by G is transitive, then R is a local ring and $J^2 = (0)$ where J is the Jacobson radical of R. In addition, if G is a simple group, then R is isomorphic to $Z_2 [x]/(x^2 + 1) or Z_4$.