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

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.

MSC:

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