On the arithmetic of polynomial ideals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914377090727936 |
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| author | Bogdanovic, Nikola Cossu, Laura Khadam, Azeem |
| author_facet | Bogdanovic, Nikola Cossu, Laura Khadam, Azeem |
| contents | This paper investigates atomic factorizations in the monoid $\mathcal I(R)$ of nonzero ideals of a multivariate polynomial ring $R$, under ideal multiplication. Building on recent advances in factorization theory for unit-cancellative monoids, we extend techniques from the paper [Geroldinger and Khadam, Ark. Mat. 60 (2022), 67-106] to construct new families of atoms in $\mathcal I(R)$, leading to a deeper understanding of its arithmetic. We further analyze the submonoid $\mathcal M\rm{on}(R)$ of monomial ideals, deriving arithmetic properties and computing sets of lengths for specific classes of ideals. The results advance the extensive study of ideal monoids within a classical algebraic framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_24455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the arithmetic of polynomial ideals Bogdanovic, Nikola Cossu, Laura Khadam, Azeem Commutative Algebra 13A15, 13B25, 13F05, 13F20, 20M12, 20M13 This paper investigates atomic factorizations in the monoid $\mathcal I(R)$ of nonzero ideals of a multivariate polynomial ring $R$, under ideal multiplication. Building on recent advances in factorization theory for unit-cancellative monoids, we extend techniques from the paper [Geroldinger and Khadam, Ark. Mat. 60 (2022), 67-106] to construct new families of atoms in $\mathcal I(R)$, leading to a deeper understanding of its arithmetic. We further analyze the submonoid $\mathcal M\rm{on}(R)$ of monomial ideals, deriving arithmetic properties and computing sets of lengths for specific classes of ideals. The results advance the extensive study of ideal monoids within a classical algebraic framework. |
| title | On the arithmetic of polynomial ideals |
| topic | Commutative Algebra 13A15, 13B25, 13F05, 13F20, 20M12, 20M13 |
| url | https://arxiv.org/abs/2510.24455 |