Galois-azumaya Extensions and the Brauer-galois Group of a Commutative Ring

Introduction. Galois extensions of noncommutative rings were introduced in 1964 by Teruo Kanzaki [13]. These algebraic objects generalize to noncommutative rings the classical Galois extensions of fields and the Galois extensions of commutative rings due to Auslander and Goldman [1]. At the same time they also turn out to be fundamental examples of Hopf-Galois extensions; these were first considered by Kreimer-Takeuchi [18] as a noncommutative analogue of the torsors in algebraic geometry. Since Galois extensions are separable (Corollary 2.4) and since the class of central Galois extensions ψ : R −→ S over a fixed commutative ground ring R behaves well under tensor product (Theorem 3.1), we may introduce a subgroup of the Brauer group of R, that we designate by Brauer-Galois group of R. The purpose of the present paper is to compute this object in some particular cases.