On separable abelian extensions of rings

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).