On the primitive representations of the group of units of a ring to the symmetric group of the nonunits

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 the regular action of G on X is faithful if and only if A is not a local ring or A is a local ring with Jacobson radical J and the left annihilator of J is the zero ideal. If X has more than one element and G acts transitively on X, then G is primitive on X if X has no nontrivial partition which is permuted by the action of G on X. It is shown that if | X| ≥ 2, then G acts primitively on X by the regular action if and only if A is isomorphic to Z (9) , to the Galois ring GR(2 2, n) where 2 n − 1 is prime or to the ring F[t;σ] (t 2) of σ-dual numbers over the field F where F = GF(3) or F = GF(2 n ) for some positive integer n such that 2 n − 1 is prime. Furthermore, a characterization of those rings for which G acts primitively on X by conjugation is given. G acts semi-primitively on X if for all x in X, Orb( x) = { x} or G acts primitively on Orb( x). A characterization of those local rings for which G acts semi-primitively on X by the regular action is given. The above results are then applied to compact topological rings.