On characterizations of a center Galois extension

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...

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Main Authors: George Szeto, Lianyong Xue
Format: Article
Language:English
Published: Wiley 2000-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Subjects:
Online Access:http://dx.doi.org/10.1155/S0161171200003562
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author George Szeto
Lianyong Xue
author_facet George Szeto
Lianyong Xue
author_sort George Szeto
collection DOAJ
description 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.
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spelling doaj-art-ced34f864f7c46a4a4e0c1706fd9571c2025-02-03T05:44:58ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252000-01-01231175375810.1155/S0161171200003562On characterizations of a center Galois extensionGeorge Szeto0Lianyong Xue1Department of Mathematics, Bradley University, Peoria 61625, Illinois, USADepartment of Mathematics, Bradley University, Peoria 61625, Illinois, USALet 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.http://dx.doi.org/10.1155/S0161171200003562Galois extensionscenter Galois extensionscentral extensions Galois central extensionsAzumaya algebrasseparable extensionsH-separable extensions.
spellingShingle George Szeto
Lianyong Xue
On characterizations of a center Galois extension
International Journal of Mathematics and Mathematical Sciences
Galois extensions
center Galois extensions
central extensions
Galois central extensions
Azumaya algebras
separable extensions
H-separable extensions.
title On characterizations of a center Galois extension
title_full On characterizations of a center Galois extension
title_fullStr On characterizations of a center Galois extension
title_full_unstemmed On characterizations of a center Galois extension
title_short On characterizations of a center Galois extension
title_sort on characterizations of a center galois extension
topic Galois extensions
center Galois extensions
central extensions
Galois central extensions
Azumaya algebras
separable extensions
H-separable extensions.
url http://dx.doi.org/10.1155/S0161171200003562
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