Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings
We determine the Dedekind domain pairs of rings; that is, pairs of rings R⊂S such that each intermediary ring in between R and S is a Dedekind domain. We also establish that if R⊂S is an extension of rings having only one non-Dedekind intermediary ring, then necessarily R is not Dedekind and so R is...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2022/4642508 |
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| author | Naseam Al-Kuleab Noômen Jarboui |
| author_facet | Naseam Al-Kuleab Noômen Jarboui |
| author_sort | Naseam Al-Kuleab |
| collection | DOAJ |
| description | We determine the Dedekind domain pairs of rings; that is, pairs of rings R⊂S such that each intermediary ring in between R and S is a Dedekind domain. We also establish that if R⊂S is an extension of rings having only one non-Dedekind intermediary ring, then necessarily R is not Dedekind and so R is a maximal non-Dedekind domain subring of S. Maximal non-Dedekind domain subrings R of S are identified in the following cases: (1) R is not integrally closed, (2) R is integrally closed and either SuppS/R<∞ or MaxR<∞, (3) S is a field, (4) R is a valuation domain, and (5) R⊂S is an integral extension. We also provide some classifications of pairs of rings having exactly two non-Dedekind domain intermediary rings. |
| format | Article |
| id | doaj-art-93682320d37a4feb9b6a1de560cd57e1 |
| institution | Kabale University |
| issn | 2314-4785 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-93682320d37a4feb9b6a1de560cd57e12025-02-03T06:07:34ZengWileyJournal of Mathematics2314-47852022-01-01202210.1155/2022/4642508Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary RingsNaseam Al-Kuleab0Noômen Jarboui1King Faisal UniversityUniversité de SfaxWe determine the Dedekind domain pairs of rings; that is, pairs of rings R⊂S such that each intermediary ring in between R and S is a Dedekind domain. We also establish that if R⊂S is an extension of rings having only one non-Dedekind intermediary ring, then necessarily R is not Dedekind and so R is a maximal non-Dedekind domain subring of S. Maximal non-Dedekind domain subrings R of S are identified in the following cases: (1) R is not integrally closed, (2) R is integrally closed and either SuppS/R<∞ or MaxR<∞, (3) S is a field, (4) R is a valuation domain, and (5) R⊂S is an integral extension. We also provide some classifications of pairs of rings having exactly two non-Dedekind domain intermediary rings.http://dx.doi.org/10.1155/2022/4642508 |
| spellingShingle | Naseam Al-Kuleab Noômen Jarboui Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings Journal of Mathematics |
| title | Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings |
| title_full | Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings |
| title_fullStr | Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings |
| title_full_unstemmed | Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings |
| title_short | Characterizations of Pairs of Rings with Few Non-Dedekind Intermediary Rings |
| title_sort | characterizations of pairs of rings with few non dedekind intermediary rings |
| url | http://dx.doi.org/10.1155/2022/4642508 |
| work_keys_str_mv | AT naseamalkuleab characterizationsofpairsofringswithfewnondedekindintermediaryrings AT noomenjarboui characterizationsofpairsofringswithfewnondedekindintermediaryrings |