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International Journal of Mathematics and Mathematical Sciences
Volume 5, Issue 4, Pages 779-784
http://dx.doi.org/10.1155/S0161171282000714

On separable abelian extensions of rings

Mathematics Department, Bradley University, Peoria 61625, Illinois, USA

Received 9 February 1982

Copyright © 1982 Hindawi Publishing Corporation. 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={cinC/(c)ρi=cfor eachρiinG}) such that R[x1,,xm]RGKC[x1,,xm] if and only if C is Galois over K with Galois group G (the Kanzaki hypothesis).