On Azumaya algebras with a finite automorphism group

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 B=(⊕∑i=1mBfi)⊕D where Bfi is a central Galois algebra with Galois group Hi|Bfi≅Hi for each i=1,2,…,m and D is contained in C.