On finite factorization Puiseux algebras
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918057939566592 |
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| author | Benelmekki, Mohamed |
| author_facet | Benelmekki, Mohamed |
| contents | An integral domain $D$ is called a finite factorization domain (FFD) if every nonzero nonunit element of $D$ has only finitely many non-associate divisors. In 1998, for an integral domain $D$ and a cancellative torsion-free monoid $S$ such that each nonzero element of its quotient group is of type $(0,0, \ldots)$, Kim proved that the monoid domain $D[S]$ is an FFD if and only if $D$ is an FFD and $S$ is an FFM. However, it is still open whether a monoid algebra $K[S]$ is an FFD provided that $S$ is a reduced FFM. In this paper, we show that a Puiseux algebra $K[S]$ is an FFD if and only if $S$ is an FFM, when $K$ is a finitely generated field of characteristic $0$. This would provide a large class of one-dimensional monoid algebras with finite factorization property. We also prove that every generalized cyclotomic polynomial has the finite factorization property in $K[S]$ where $S$ is a reduced FFM and $K$ is an arbitrary field of characteristic $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On finite factorization Puiseux algebras Benelmekki, Mohamed Commutative Algebra 13A05, 13F15, 13F25, 13G05 An integral domain $D$ is called a finite factorization domain (FFD) if every nonzero nonunit element of $D$ has only finitely many non-associate divisors. In 1998, for an integral domain $D$ and a cancellative torsion-free monoid $S$ such that each nonzero element of its quotient group is of type $(0,0, \ldots)$, Kim proved that the monoid domain $D[S]$ is an FFD if and only if $D$ is an FFD and $S$ is an FFM. However, it is still open whether a monoid algebra $K[S]$ is an FFD provided that $S$ is a reduced FFM. In this paper, we show that a Puiseux algebra $K[S]$ is an FFD if and only if $S$ is an FFM, when $K$ is a finitely generated field of characteristic $0$. This would provide a large class of one-dimensional monoid algebras with finite factorization property. We also prove that every generalized cyclotomic polynomial has the finite factorization property in $K[S]$ where $S$ is a reduced FFM and $K$ is an arbitrary field of characteristic $0$. |
| title | On finite factorization Puiseux algebras |
| topic | Commutative Algebra 13A05, 13F15, 13F25, 13G05 |
| url | https://arxiv.org/abs/2506.11793 |