Some correspondences for Galois skew group rings. (English) Zbl 0990.16025

Szeto, G.; Xue, L.

Port. Math. 57, No. 2, 161-168 (2000).

Let \(G\) be a finite automorphism group of a ring \(S\) with identity such that the order of \(G\) is invertible in \(S\). Suppose that the skew group ring \(S*G\) of \(G\) over \(S\) is an Azumaya \(Z\)-algebra, where \(Z\) is the center of \(S*G\). Let \(G'\) be the inner automorphism group of \(S*G\) induced by the elements of \(G\). When \(S*G\) is a \(G'\)-Galois extension, the authors define two equivalence relations in the set of subgroups of \(G\) and prove two correspondence theorems between the classes of equivalent subgroups and a class of the Azumaya algebras contained in \(S*G\). Moreover, they show that \(S*G\) is a composition of three Galois extensions, and give an expression of \(Z\) and the fixed subring \((S*G)^{G'}\) under \(G'\) when \(G\) is Abelian.

Reviewer: S.V.Mihovski (Plovdiv)

MSC:

16S35	Twisted and skew group rings, crossed products
16H05	Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16W20	Automorphisms and endomorphisms

Keywords:

finite automorphism groups; skew group rings; centers; inner automorphisms; Azumaya algebras; Galois extensions; fixed subrings