If R is a ring with identity and X is the set of all nonzero, nonunits of Rand G is the group of all units of R, then naturally, the three group actions

are considerted as follows; For all

and all

, left regular action

, right regular action

and conjugate action

. For

, let 0(x) denote the orbit of x under the given action of G on X. G is transitive on X if X is 0(x) for some

. G is half-transitive on X if G is transitive on X or if 0(x) is a finite set with |0(x)| > 1 and |0(x)| = |0(x)| for all x and

. A characterization of those left Artinian ring with identity for which G is half-transitive on X by the left (and right) regular action is given, and as a corollary, we have that in a left Artinian ring with identity, G is half-transitive on X by the left regular action if and only if G is half-transitive on X by the right regular action. It is shown that if R is semisimple left Artinian ring, then 0(x), the orbit of x under the conjugation action, is a finite set and |0(x)| = |0(x)| for all x and

if and only if R is isomorphic to a direct sum of finite number of fields, and as a corollary, we know that there is neither half-transitive nor transitive conjugate action of G on X. It is also shown that in a left Artinian ring R with identity for which G is half-transitive on X by conjugate action, if the characteristic of R is finite, then the characteristic of R is a prime p, and moreover if |0(x)| is relatively prime to p for some

, then R is a local ring.