Non-commutative separability and group actions

Abstract

We give conditions for the skew group ring S * G to be strongly separable and H-separable over the ring S. In particular we show that the H-separability is equivalent to S being central Galois extension. We also look into the H-separability of the ring S over the fixed subring R under a faithful action of a group G. We show that such a chain: S * G H-separable over S and S H-separable over R cannot occur, and that the centralizer of R in S is an Azumaya algebra in the presente of a central element of trace one.In [A] we introduced the concept of subring-Galois extensions as a generalization of central Galois extensions and give a generalization of the correspondence theorem given by DeMeyer in [D] and Szeto in [SM] . Similar correspondence theorems were given by Sugano in [S] using Hseparability. Separability for non-commutative rings was introduced by Hirata, and the notions of H-separability and "strong" separability were introduced by Hirata in [HI] and MacMahon and Mewborn in [MM] respectively . Strong separability is a weaker notion than H-separability, but both are special cases of the general notion of separability of ring extensions.In the case of group actions we present here conditions for strong and H-separability of skew group rings and in particular we show that the skew group ring S * G is H-separable over S if and only if S is a central Galois extension . Furthermore, in this case S*G is a Z(S)-Galois