Integrality Over Fixed Rings

Abstract

In this paper we prove that if G is a finite group of automorphisms of a ring R, where the order of G is a unit in R, then R is fully integral of degree m(|G|) over the fixed ring RG. Here m is a function of the order of G. This gives a positive answer to a well-known question of S. Montgomery and extends the result of D. S. Passman for abelian group actions. Our theorem can be viewed as a generalization of the Bergman–Isaacs theorem. In fact that result can be obtained as a corollary, albeit with a poorer index of nilpotence. We then briefly consider duality for Hopf algebra actions and conclude by proving an integrality result for an inner action by a finite-dimensional semisimple Hopf algebra.