## 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
Full Text: