On GL-domains and the ascent of the IDF property
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910914977988608 |
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| author | Gonzalez, Victor Panpaliya, Ishan |
| author_facet | Gonzalez, Victor Panpaliya, Ishan |
| contents | Following the terminology introduced by Arnold and Sheldon back in 1975, we say that an integral domain $D$ is a GL-domain if the product of any two primitive polynomials over $D$ is again a primitive polynomial. In this paper, we study the class of GL-domains. First, we propose a characterization of GL-domain in terms of certain elements we call prime-like. Then we identify a new class of GL-domains. An integral domain $D$ is also said to have the IDF property provided that each nonzero element of $D$ is divisible by only finitely many non-associate irreducible divisors. It was proved by Malcolmson and Okoh in 2009 that the IDF property ascends to polynomial extensions when restricted to the class of GCD-domains. This result was recently strengthened by Gotti and Zafrullah to the class of PSP-domains. We conclude this paper by proving that the IDF property does not ascend to polynomial extensions in the class of GL-domains, answering an open question posed by Gotti and Zafrullah. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On GL-domains and the ascent of the IDF property Gonzalez, Victor Panpaliya, Ishan Commutative Algebra Primary: 13A05, 13F15, Secondary: 13P05 Following the terminology introduced by Arnold and Sheldon back in 1975, we say that an integral domain $D$ is a GL-domain if the product of any two primitive polynomials over $D$ is again a primitive polynomial. In this paper, we study the class of GL-domains. First, we propose a characterization of GL-domain in terms of certain elements we call prime-like. Then we identify a new class of GL-domains. An integral domain $D$ is also said to have the IDF property provided that each nonzero element of $D$ is divisible by only finitely many non-associate irreducible divisors. It was proved by Malcolmson and Okoh in 2009 that the IDF property ascends to polynomial extensions when restricted to the class of GCD-domains. This result was recently strengthened by Gotti and Zafrullah to the class of PSP-domains. We conclude this paper by proving that the IDF property does not ascend to polynomial extensions in the class of GL-domains, answering an open question posed by Gotti and Zafrullah. |
| title | On GL-domains and the ascent of the IDF property |
| topic | Commutative Algebra Primary: 13A05, 13F15, Secondary: 13P05 |
| url | https://arxiv.org/abs/2504.10722 |