Skew group rings which are azumaya

Abstract

If S is a ring with 1, and G is a finite group acting faithfully as automorphisms of S, then it is well known that the skew group ring S ∗ G is a separable extension of S if and only if there exists a central element in S with trace one. A ring is an Azumaya algebra if it is separable over its center. The case of when the group ring S[G] is Azumaya was studied by De Meyer and Janusz in [2] ; and lately the case of twisted group rings was studied by Szeto and Wong in [5] ; but the techniques used on those cases cannot be applied to the skew group rings precisely because the elements of the ring S do not commute with the elements of the group G. The purpose of this paper is to give conditions on the action of the group in order to make the skew group ring an Azumaya algebra. A ring A is said to be separable over a subring B if the (A − A)-module homomorphism of A⊗BA onto A defined by a⊗b 7−→ ab splits, and A is called H-separable over B if A ⊗B A is isomorphic as (A − A)-bimodule to a direct summand of A for some n ≥ 1. Clearly H-separable extension are separable. Let S be a ring with 1 and G be a finite group acting as automorphisms of S with fixed ring S; that we will denote R. The skew group ring S ∗G is the free left S-module with basis G, where multiplication is defined according