On rings of integer-valued rational functions
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917829448564736 |
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| author | Chems-Eddin, Mohamed Mahmoud Feryouch, Badr Mouanis, Hakima Tamoussit, Ali |
| author_facet | Chems-Eddin, Mohamed Mahmoud Feryouch, Badr Mouanis, Hakima Tamoussit, Ali |
| contents | Let $D\subseteq B$ be an extension of integral domains and $E$ a subset of the quotient field of $D$. We introduce the ring of \textit{$D$-valued $B$-rational functions on $E$}, denoted by $Int^R_B(E,D)$, which naturally extends the concepts of integer-valued polynomials, defined as $ Int^R_B(E,D) \:=\lbrace f \in B(X);\; f(E)\subseteq D\rbrace.$ The notion of $Int^R_B(E,D)$ boils down to the usual notion of integer-valued rational functions when the subset $E$ is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from $Int^R(E,D)$ to $D$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_16142 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On rings of integer-valued rational functions Chems-Eddin, Mohamed Mahmoud Feryouch, Badr Mouanis, Hakima Tamoussit, Ali Commutative Algebra Let $D\subseteq B$ be an extension of integral domains and $E$ a subset of the quotient field of $D$. We introduce the ring of \textit{$D$-valued $B$-rational functions on $E$}, denoted by $Int^R_B(E,D)$, which naturally extends the concepts of integer-valued polynomials, defined as $ Int^R_B(E,D) \:=\lbrace f \in B(X);\; f(E)\subseteq D\rbrace.$ The notion of $Int^R_B(E,D)$ boils down to the usual notion of integer-valued rational functions when the subset $E$ is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from $Int^R(E,D)$ to $D$. |
| title | On rings of integer-valued rational functions |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2410.16142 |