George Szeto

Copyright © 1982 George Szeto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let R be a ring with 1, G(=〈ρ1〉×…×〈ρm〉) a finite abelian automorphism group of R of order n where 〈ρi〉 is cyclic of order ni. for some integers n, ni, and m, and C the center of R whose automorphism group induced by G is isomorphic with G. Then an abelian extension R[x1,…,xm] is defined as a generalization of cyclic extensions of rings, and R[x1,…,xm] is an Azumaya algebra over K(=CG={c   in   C/(c)ρi=c   for each   ρi   in   G}) such that R[x1,…,xm]≅RG⊗KC[x1,…,xm] if and only if C is Galois over K with Galois group G (the Kanzaki hypothesis).