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arXiv · 2410.16142

On rings of integer-valued rational functions

Mohamed Mahmoud Chems-Eddin·Badr Feryouch·Hakima Mouanis·Ali Tamoussit
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Abstract

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 $D$.

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Mohamed Mahmoud Chems-Eddin, Badr Feryouch, Hakima Mouanis, Ali Tamoussit. 2024-10-21. On rings of integer-valued rational functions. https://arxiv.org/abs/2410.16142

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