On Azumaya algebras with a finite automorphism group

George Szeto,Lianyong Xue

doi:10.1155/s0161171201006068

George Szeto, Lianyong Xue

Open Access

https://doi.org/10.1155/s0161171201006068

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Abstract

Let B be a ring with 1, C the center of B, and G a finite automorphism group of B. It is shown that if B is an Azumaya algebra such that B = ⊕∑g∈GJg where Jg = {b ∈ B|bx = g(x)b for all x ∈ B}, then there exist orthogonal central idempotents {fi ∈ C|i = 1, 2, …, m for some integer m} and subgroups Hi of G such that where Bfi is a central Galois algebra with Galois group for each i = 1, 2, …, m and D is contained in C.

Highlights

Let A be an Azumaya algebra, G a finite algebra automorphism group of A, and Jg = {a ∈ A | ax = g(x)a for all x ∈ A} for each g ∈ G
In [6], it was shown that JgJh = Jgh for all g, h ∈ G
In [2], let B be a separable algebra over a commutative ring R and G a finite algebra automorphism group of B

Summary

Introduction

Let A be an Azumaya algebra, G a finite algebra automorphism group of A, and Jg = {a ∈ A | ax = g(x)a for all x ∈ A} for each g ∈ G. In [2], let B be a separable algebra over a commutative ring R and G a finite algebra automorphism group of B. B is a central Galois algebra with Galois group G if and only if for each g ∈ G, JgJg−1 = C, the center of B.

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