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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| Format: | Article |
| Language: | English |
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Wiley
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171200003562 |
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| _version_ | 1832556572628221952 |
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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. |
| format | Article |
| id | doaj-art-ced34f864f7c46a4a4e0c1706fd9571c |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2000-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| 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 |
| work_keys_str_mv | AT georgeszeto oncharacterizationsofacentergaloisextension AT lianyongxue oncharacterizationsofacentergaloisextension |