On generalized quaternion algebras
Let B be a commutative ring with 1, and G(={σ}) an automorphism group of B of order 2. The generalized quaternion ring extension B[j] over B is defined by S. Parimala and R. Sridharan such that (1) B[j] is a free B-module with a basis {1,j}, and (2) j2=−1 and jb=σ(b)j for each b in B. The purpose of...
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| Format: | Article |
| Language: | English |
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Wiley
1980-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171280000166 |
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| _version_ | 1832548614048579584 |
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| author | George Szeto |
| author_facet | George Szeto |
| author_sort | George Szeto |
| collection | DOAJ |
| description | Let B be a commutative ring with 1, and G(={σ}) an automorphism group of B of order 2. The generalized quaternion ring extension B[j] over B is defined by S. Parimala and R. Sridharan such that (1) B[j] is a free B-module with a basis {1,j}, and (2) j2=−1 and jb=σ(b)j for each b in B. The purpose of this paper is to study the separability of B[j]. The separable extension of B[j] over B is characterized in terms of the trace (=1+σ) of B over the subring of fixed elements under σ. Also, the characterization of a Galois extension of a commutative ring given by Parimala and Sridharan is improved. |
| format | Article |
| id | doaj-art-79b73c7994ed469f949c2521cfa1e56a |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1980-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-79b73c7994ed469f949c2521cfa1e56a2025-02-03T06:13:39ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251980-01-013223724510.1155/S0161171280000166On generalized quaternion algebrasGeorge Szeto0Department of Mathematics, Bradley University, Peoria 61625, Illinois, USALet B be a commutative ring with 1, and G(={σ}) an automorphism group of B of order 2. The generalized quaternion ring extension B[j] over B is defined by S. Parimala and R. Sridharan such that (1) B[j] is a free B-module with a basis {1,j}, and (2) j2=−1 and jb=σ(b)j for each b in B. The purpose of this paper is to study the separability of B[j]. The separable extension of B[j] over B is characterized in terms of the trace (=1+σ) of B over the subring of fixed elements under σ. Also, the characterization of a Galois extension of a commutative ring given by Parimala and Sridharan is improved.http://dx.doi.org/10.1155/S0161171280000166quaternion ringsseparable algebrasand Galois extensions. |
| spellingShingle | George Szeto On generalized quaternion algebras International Journal of Mathematics and Mathematical Sciences quaternion rings separable algebras and Galois extensions. |
| title | On generalized quaternion algebras |
| title_full | On generalized quaternion algebras |
| title_fullStr | On generalized quaternion algebras |
| title_full_unstemmed | On generalized quaternion algebras |
| title_short | On generalized quaternion algebras |
| title_sort | on generalized quaternion algebras |
| topic | quaternion rings separable algebras and Galois extensions. |
| url | http://dx.doi.org/10.1155/S0161171280000166 |
| work_keys_str_mv | AT georgeszeto ongeneralizedquaternionalgebras |