International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 11, Pages 753-758
doi:10.1155/S0161171200003562
On characterizations of a center Galois extension
Department of Mathematics, Bradley University, Peoria 61625, Illinois, USA
Received 16 June 1999
Copyright © 2000 George Szeto and Lianyong Xue. 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 B be a ring with 1, C the center of B, G a finite
automorphism group of B, and BG the set of elements in B
fixed under each element in G. Then, it is shown that B is a
center Galois extension of BG (that is, C is a Galois algebra
over CG with Galois group G|C≅G) if and only if the
ideal of B generated by {c−g(c)|c∈C} is B for each
g≠1 in G. This generalizes the well known characterization
of a commutative Galois extension C that C is a Galois
extension of CG with Galois group G if and only if the ideal
generated by {c−g(c)|c∈C} is C for each g≠1 in
G. Some more characterizations of a center Galois extension B
are also given.