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On rings of integer-valued rational functions

Mohamed Mahmoud Chems-Eddin, Badr Feryouch, Hakima Mouanis, Ali Tamoussit

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 $D$.

On rings of integer-valued rational functions

Abstract

Let be an extension of integral domains and a subset of the quotient field of . We introduce the ring of \textit{-valued -rational functions on }, denoted by , which naturally extends the concepts of integer-valued polynomials, defined as The notion of boils down to the usual notion of integer-valued rational functions when the subset 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 to .

Paper Structure

This paper contains 4 sections, 50 theorems, 13 equations.

Table of Contents

  1. Preliminary definitions and properties
  2. Localization and ideals of $\mathrm{Int^R}(E,D)$
  3. The transfer of some properties in $\mathrm{Int^R}(E,D)$
  4. The module structure of $\mathrm{Int^R}(E,D)$ and $\mathrm{Int}_B(E,D)$

Key Result

Proposition 1.1

Let $D_1\subseteq D_2$ be two integral domains with the same quotient field $K$ and $E\subseteq F$ two nonempty subsets of $K$. Let $D_i\subseteq B_i$ be two extensions of integral domains for $i\in \{1,2\}$. If $B_1 \subseteq B_2$, then $\mathrm{Int}^\mathrm{R}_{B_1} \left(F,D_1\right) \subseteq \

Theorems & Definitions (88)

  • Proposition 1.1
  • Corollary 1.2
  • proof
  • Proposition 1.3
  • Lemma 1.4
  • proof
  • Proposition 1.5
  • proof
  • Proposition 1.7
  • proof
  • ...and 78 more