Other Examples of Principal Ideal Domains that are not Euclidean Domains
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912692326891520 |
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| author | Allo-Gómez, Nicolás |
| author_facet | Allo-Gómez, Nicolás |
| contents | It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic integers in a certain, very specific quadratic field, and the proof that this works is quite unnatural and technical. In this article, we will present a family of counterexamples constructed using real closed fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_17464 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Other Examples of Principal Ideal Domains that are not Euclidean Domains Allo-Gómez, Nicolás Commutative Algebra Rings and Algebras 13F07, 13F10 It is a well-known and easily established fact that every Euclidean domain is also a principal ideal domain. However, the converse statement is not true, and this is usually shown by exhibiting as a counterexample the ring of algebraic integers in a certain, very specific quadratic field, and the proof that this works is quite unnatural and technical. In this article, we will present a family of counterexamples constructed using real closed fields. |
| title | Other Examples of Principal Ideal Domains that are not Euclidean Domains |
| topic | Commutative Algebra Rings and Algebras 13F07, 13F10 |
| url | https://arxiv.org/abs/2503.17464 |