On rings of integer-valued rational functions

TL;DR

This paper introduces and studies rings of integer-valued rational functions over integral domain extensions, exploring their algebraic properties and how they relate to classical integer-valued polynomial rings.

Contribution

It extends the concept of integer-valued polynomials to rational functions over domain extensions and investigates their algebraic structure and properties.

Findings

Analysis of prime ideals and localization of the rings

Characterization of module structures over these rings

Transfer of ring-theoretic properties from classical integer-valued polynomial rings

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 $D$.