On separable abelian extensions of rings
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 general...
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| Format: | Article |
| Language: | English |
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Wiley
1982-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171282000714 |
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| _version_ | 1849413715083919360 |
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| author | George Szeto |
| author_facet | George Szeto |
| author_sort | George Szeto |
| collection | DOAJ |
| description | 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={c in C/(c)ρi=c for each ρi in G}) such that R[x1,…,xm]≅RG⊗KC[x1,…,xm] if and only if C is Galois over K with Galois group G (the Kanzaki hypothesis). |
| format | Article |
| id | doaj-art-67d0d901ed724f7583ff860dce5fdfb0 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1982-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-67d0d901ed724f7583ff860dce5fdfb02025-08-20T03:34:02ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251982-01-015477978410.1155/S0161171282000714On separable abelian extensions of ringsGeorge Szeto0Mathematics Department, Bradley University, Peoria 61625, Illinois, USALet 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={c in C/(c)ρi=c for each ρi in G}) such that R[x1,…,xm]≅RG⊗KC[x1,…,xm] if and only if C is Galois over K with Galois group G (the Kanzaki hypothesis).http://dx.doi.org/10.1155/S0161171282000714Abelian ring extensionsseparable algebrasAzumaya algebrasGalois extensions. |
| spellingShingle | George Szeto On separable abelian extensions of rings International Journal of Mathematics and Mathematical Sciences Abelian ring extensions separable algebras Azumaya algebras Galois extensions. |
| title | On separable abelian extensions of rings |
| title_full | On separable abelian extensions of rings |
| title_fullStr | On separable abelian extensions of rings |
| title_full_unstemmed | On separable abelian extensions of rings |
| title_short | On separable abelian extensions of rings |
| title_sort | on separable abelian extensions of rings |
| topic | Abelian ring extensions separable algebras Azumaya algebras Galois extensions. |
| url | http://dx.doi.org/10.1155/S0161171282000714 |
| work_keys_str_mv | AT georgeszeto onseparableabelianextensionsofrings |