Separable subalgebras of a class of Azumaya algebras

George Szeto

doi:10.1155/s0161171298000337

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 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.

Highlights

DeMeyer and Kanzaki [2] studied central Galois algebras and Galois extensions whose center is a Galois algebra with Galois group induced by and isomorphic with the group of the extension. These two types of Galois extensions were recently generalized to a bgger class of Galois Azumaya extenmons [3]
Where S is called a G-Galois Azumaya extension of SG if S is a G-Galois extension of SG that is an Azumaya CG-algebra where C is the center of S and SG IS the subring of elements fixed under each element of G
Sugano [4] investigated a G-Galois H-separable extension of S and recently, Szeto [5] proved that a G-Galos H-separable extension S of SG that is a projective separable CG-algebra if and only f S is a CG-Azumaya algebra

Summary

INTRODUCTION

DeMeyer and Kanzaki [2] studied central Galois algebras and Galois extensions whose center is a Galois algebra with Galois group induced by and isomorphic with the group of the extension These two types of Galois extensions were recently generalized to a bgger class of Galois Azumaya extenmons [3]. Sugano [4] investigated a G-Galois H-separable extension of S and recently, Szeto [5] proved that a G-Galos H-separable extension S of SG that is a projective separable CG-algebra if and only f S is a CG-Azumaya algebra. We call such an S a GHS-extension. We call (R){ci, di} a G-Galois system for S

SKEW GROUP RINGS

SEPARABLE ALGEBRAS