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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)
MSC:
 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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