The group of units 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 the additive group A + of A by left multiplication (the regular action) and by conjugation (the conjugate action). If X is the set of nonzero, nonunits of A, then X is invariant under both actions. It is shown that if A is a compact ring with identity and X is the union of finitely many orbits by the regular action or the conjugate action, then A is finite. Moreover, a characterization of those compact rings with identity for which either action is transitive on X is given. In a compact ring A with identity, the structure of A is reflected by that of the group of units of A. It is shown that if 2 is a unit in A, then A is finite if and only if G is finite and A is commutative if and only if G is abelian. Moreover, if the conjugate action on X is trivial, then A is a commutative ring.