Control subgroups and birational extensions of graded rings
In this paper, we establish the relation between the concept of control subgroups and the class of graded birational algebras. Actually, we prove that if R=⊕σ∈GRσ is a strongly G-graded ring and H⊲G, then the embedding i:R(H)↪R, where R(H)=⊕σ∈HRσ, is a Zariski extension if and only if H controls the...
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| Format: | Article |
| Language: | English |
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Wiley
1999-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171299224118 |
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| _version_ | 1841524526695841792 |
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| author | Salah El Din S. Hussein |
| author_facet | Salah El Din S. Hussein |
| author_sort | Salah El Din S. Hussein |
| collection | DOAJ |
| description | In this paper, we establish the relation between the concept of control subgroups and the class of graded birational algebras. Actually, we prove that if R=⊕σ∈GRσ is a strongly G-graded ring and H⊲G, then the embedding i:R(H)↪R, where R(H)=⊕σ∈HRσ, is a Zariski extension if and only if H controls the filter ℒ(R−P) for every prime ideal P in an open set of the Zariski topology on R. This enables us to relate certain ideals of R and R(H) up to radical. |
| format | Article |
| id | doaj-art-36ebde8c07144a888fb404f97a771ce7 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1999-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-36ebde8c07144a888fb404f97a771ce72025-02-03T05:52:55ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251999-01-0122241141510.1155/S0161171299224118Control subgroups and birational extensions of graded ringsSalah El Din S. Hussein0Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo 11566, EgyptIn this paper, we establish the relation between the concept of control subgroups and the class of graded birational algebras. Actually, we prove that if R=⊕σ∈GRσ is a strongly G-graded ring and H⊲G, then the embedding i:R(H)↪R, where R(H)=⊕σ∈HRσ, is a Zariski extension if and only if H controls the filter ℒ(R−P) for every prime ideal P in an open set of the Zariski topology on R. This enables us to relate certain ideals of R and R(H) up to radical.http://dx.doi.org/10.1155/S0161171299224118Control subgroupsbirational extensionsZariski extensionsGabriel filterskernel functors. |
| spellingShingle | Salah El Din S. Hussein Control subgroups and birational extensions of graded rings International Journal of Mathematics and Mathematical Sciences Control subgroups birational extensions Zariski extensions Gabriel filters kernel functors. |
| title | Control subgroups and birational extensions of graded rings |
| title_full | Control subgroups and birational extensions of graded rings |
| title_fullStr | Control subgroups and birational extensions of graded rings |
| title_full_unstemmed | Control subgroups and birational extensions of graded rings |
| title_short | Control subgroups and birational extensions of graded rings |
| title_sort | control subgroups and birational extensions of graded rings |
| topic | Control subgroups birational extensions Zariski extensions Gabriel filters kernel functors. |
| url | http://dx.doi.org/10.1155/S0161171299224118 |
| work_keys_str_mv | AT salaheldinshussein controlsubgroupsandbirationalextensionsofgradedrings |