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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2401.11602 |
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| _version_ | 1866910304543178752 |
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| author | Korbelář, Miroslav |
| author_facet | Korbelář, Miroslav |
| contents | Let $P$ be a finitely generated commutative semiring. It was shown recently that if $P$ is a parasemifield (i.e. the multiplicative reduct of $P$ is a group) then $P$ cannot contain the positive rationals $\mathbb{Q}^+$ as its subsemiring. Equivalently, a commutative parasemifield $P$ finitely generated as a semiring is additively divisible if and only if $P$ is additively idempotent.
We generalize this result using weaker forms of these additive properties to a broader class of commutative semirings in the following way.
Let $S$ be a semiring that is a factor of a monoid semiring $\mathbb{N}[\mathcal{C}]$ where $\mathcal{C}$ is a submonoid of a free commutative monoid of finite rank. Then the semiring $S$ is additively almost-divisible if and only if $S$ is torsion. In particular, we show that if $S$ is a ring then $S$ cannot contain any non-finitely generated subring of $\mathbb{Q}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11602 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torsion factors of commutative monoid semirings Korbelář, Miroslav Rings and Algebras Let $P$ be a finitely generated commutative semiring. It was shown recently that if $P$ is a parasemifield (i.e. the multiplicative reduct of $P$ is a group) then $P$ cannot contain the positive rationals $\mathbb{Q}^+$ as its subsemiring. Equivalently, a commutative parasemifield $P$ finitely generated as a semiring is additively divisible if and only if $P$ is additively idempotent. We generalize this result using weaker forms of these additive properties to a broader class of commutative semirings in the following way. Let $S$ be a semiring that is a factor of a monoid semiring $\mathbb{N}[\mathcal{C}]$ where $\mathcal{C}$ is a submonoid of a free commutative monoid of finite rank. Then the semiring $S$ is additively almost-divisible if and only if $S$ is torsion. In particular, we show that if $S$ is a ring then $S$ cannot contain any non-finitely generated subring of $\mathbb{Q}$. |
| title | Torsion factors of commutative monoid semirings |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2401.11602 |