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On Azumaya Galois extensions and skew group rings. (English) Zbl 0926.16026
Let \(S\) be a ring with 1 and let \(G\) be a finite subgroup of automorphisms of \(S\). The author gives two equivalent definitions of what he calls an Azumaya Galois extension. In a previous paper the author defines \(S\) to be an Azumaya Galois extension if \(S/S^G\) is Galois and \(S^G\) is an Azumaya \(C^G\)-algebra, where \(C\) is the center of \(S\), and shows that this condition is equivalent to the skew group ring \(S*G\) being an Azumaya \(C^G\)-algebra. Here the author gives two more equivalent conditions to being an Azumaya Galois extension. First, the skew group ring \(S*G\) is Azumaya over its center and the ring \(S\) satisfies the double centralizer property in \(S*G\). Second, the skew group ring is Azumaya over its center \(Z\), is also a Galois extension under the inner action induced by the group \(G\) on the skew group ring, and the group ring \(ZG\) is a finitely generated and projective \(C^G\)-module of rank \(| G|\).
For a normal subgroup \(H\) of \(G\), let \(K\) be the commutator of \(H\) in \(G\). The author also shows that the extension \((S*G)^K/(S*G)^G\) by the group \((G/K)\) is a Galois extension if and only if the \(H\)-trace of every element of the group \(G\) not in \(K\) is zero. Furthermore, in this case the author gives a Galois system.
Reviewer: R.Alfaro (Flint)
16S35 Twisted and skew group rings, crossed products
16W20 Automorphisms and endomorphisms
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
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