Half-transitive group actions in a compact ring

Abstract

If A is a ring with identity and G is the group of units of A, then G acts naturally on A +, the additive group of A, by left multiplication (the regular action) and by conjugation (the conjugate action). Clearly the set X of nonzero, nonunits of A is invariant under both actions. For x ∈ X, let O( x) denote the orbit of x under the given action of G. G is half-transitive on X if G is transitive on X or if O( x) is a finite set with cardinality greater than one for each x in X and | O( x)| = | O( y)| for all x and y in X. A characterization of those compact rings with identity for which G is half- transitive on X by the regular action is given. It is also shown that if A is a compact ring with identity such that G is half-transitive, but not transitive, on X by the conjugate action, then the characteristic of A is a prime p and | O( x)| ≥3 for all x in X. Moreover, A is a local ring if and only if (| O( x)|, p) = 1.