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).