When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring
Let R⊂S be an extension of commutative rings, with X an indeterminate, such that the extension RX⊂SX of Nagata rings has FIP (i.e., SX has only finitely many RX-subalgebras). Then, the number of RX-subalgebras of SX equals the number of R-subalgebras of S. In fact, the function from the set of R-sub...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2014/315919 |
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| author | David E. Dobbs Gabriel Picavet Martine Picavet-L’Hermitte |
| author_facet | David E. Dobbs Gabriel Picavet Martine Picavet-L’Hermitte |
| author_sort | David E. Dobbs |
| collection | DOAJ |
| description | Let R⊂S be an extension of commutative rings, with X an indeterminate, such that the extension RX⊂SX of Nagata rings has FIP (i.e., SX has only finitely many RX-subalgebras). Then, the number of RX-subalgebras of SX equals the number of R-subalgebras of S. In fact, the function from the set of R-subalgebras of S to the set of RX-subalgebras of SX given by T ↦TX is an order-isomorphism. |
| format | Article |
| id | doaj-art-7de03826e3414486baef63160ebb7632 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-7de03826e3414486baef63160ebb76322025-02-03T01:01:18ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252014-01-01201410.1155/2014/315919315919When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata RingDavid E. Dobbs0Gabriel Picavet1Martine Picavet-L’Hermitte2Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USALaboratoire de Mathématiques, Université Blaise Pascal, UMR6620, CNRS, Les Cézeaux, 24 avenue des Landais, BP 80026, 63177 Aubière Cedex, FranceLaboratoire de Mathématiques, Université Blaise Pascal, UMR6620, CNRS, Les Cézeaux, 24 avenue des Landais, BP 80026, 63177 Aubière Cedex, FranceLet R⊂S be an extension of commutative rings, with X an indeterminate, such that the extension RX⊂SX of Nagata rings has FIP (i.e., SX has only finitely many RX-subalgebras). Then, the number of RX-subalgebras of SX equals the number of R-subalgebras of S. In fact, the function from the set of R-subalgebras of S to the set of RX-subalgebras of SX given by T ↦TX is an order-isomorphism.http://dx.doi.org/10.1155/2014/315919 |
| spellingShingle | David E. Dobbs Gabriel Picavet Martine Picavet-L’Hermitte When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring International Journal of Mathematics and Mathematical Sciences |
| title | When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring |
| title_full | When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring |
| title_fullStr | When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring |
| title_full_unstemmed | When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring |
| title_short | When an Extension of Nagata Rings Has Only Finitely Many Intermediate Rings, Each of Those Is a Nagata Ring |
| title_sort | when an extension of nagata rings has only finitely many intermediate rings each of those is a nagata ring |
| url | http://dx.doi.org/10.1155/2014/315919 |
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