Separable subalgebras of a class of Azumaya algebras

Abstract

Let S be a ring with 1, C the center of S, G a finite automorphism group of S of order n invertible in S, and SG the subnng of elements of S fixed under each element in G. It is shown that the skew group ring S*G is a G′-Galois extension of (S*G)G′ that is a projective separable CG-algebra where G′ is the inner automorphism group of S*G induced by G if and only if S is a G-Galois extension of SG that is a projective separable CG-algebra. Moreover, properties of the separable subalgebras of a G-Galois H-separable extension S of SG are given when SG is a projective separable CG-algebra.

Keywords

• Azumaya algebras
• Galois extensions
• H-separable extensions
• Skew group tings.