On S-Integral Domains and S-Version of Krull Intersection Theorem
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911335150780416 |
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| author | Singh, Tushar Verma, Gyanendra K. Kumar, Shiv Datt |
| author_facet | Singh, Tushar Verma, Gyanendra K. Kumar, Shiv Datt |
| contents | Let $S\subseteq R$ be a multiplicatively closed subset of a ring $R$. We extend several results on integral domains to their $S$-versions and establish the $S$-version of Krull intersection theorem. We also show that if $R$ is an $S$-field, then the localization of $R$ with respect to $S$ is a $ϕ(S)$-field, where $ϕ(S)=\left \{\dfrac{s}{1}| \ s\in S\right \}$ is a multiplicatively closed subset of $S^{-1}R$, and prove the converse under the condition of finiteness of $S$. As a consequence, we show that every finite $S$-integral domain is an $S$-field. Also, we provide several examples to illustrate the significance of our findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_20316 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On S-Integral Domains and S-Version of Krull Intersection Theorem Singh, Tushar Verma, Gyanendra K. Kumar, Shiv Datt Commutative Algebra 12E20, 13G05, 16U10, 16U40 Let $S\subseteq R$ be a multiplicatively closed subset of a ring $R$. We extend several results on integral domains to their $S$-versions and establish the $S$-version of Krull intersection theorem. We also show that if $R$ is an $S$-field, then the localization of $R$ with respect to $S$ is a $ϕ(S)$-field, where $ϕ(S)=\left \{\dfrac{s}{1}| \ s\in S\right \}$ is a multiplicatively closed subset of $S^{-1}R$, and prove the converse under the condition of finiteness of $S$. As a consequence, we show that every finite $S$-integral domain is an $S$-field. Also, we provide several examples to illustrate the significance of our findings. |
| title | On S-Integral Domains and S-Version of Krull Intersection Theorem |
| topic | Commutative Algebra 12E20, 13G05, 16U10, 16U40 |
| url | https://arxiv.org/abs/2512.20316 |