On galois extensions of an azumaya algebra

Abstract

F. R. DeMeyer ([3]), T. Kanzaki ([7]) and M. Harada ([5]) investigated central Galois Algebras, i.e., Galois algebras Λ over k with group G such that k is the center of Λ. The present authors in [1] generalized the class of central Galois algebras to the class of Galois Azumaya extensions (Galois extensions of an Azumaya algebra), where they characterized such extensions in terms of the induced skew group ring being an Azumaya algebra or being H–separable over the ground ring. In the same paper, the authors gave a description of the ring S as a tensor product of fixed ring and the corresponding commutator subring, which is a central Galois algebra. In the present paper, we shall show two one-to-one correspondence theorems for a Galois Azumaya extension. One of the correspondence is between two sets of separable subextensions, with one set coming from the Azumaya algebra; and the other correspondence is a generalization of the correspondence given by DeMeyer in [3] for separable algebras whose centers are Galois extensions. Moreover, the structure theorem of DeMeyer for central Galois algebras whith an inner Galois group, and the characterization of Kanzaki-Harada of central Galois algebras are generalized to a Galois Azumaya extension.