Ring Structure of Integer-Valued Rational Functions

TL;DR

This paper investigates the algebraic structure of integer-valued rational functions over domains, characterizing conditions under which these rings are Pr"ufer or Be9zout domains, extending known classifications.

Contribution

It provides new characterizations of when the ring of integer-valued rational functions over a valuation domain or a general domain is Pr"ufer or Be9zout, advancing the understanding of their algebraic properties.

Findings

Characterization of when a0Int^R(V) is Prfcer domain

Conditions for a0Int^R(V) to be Be9zout domain

Extension of classification for a0Int^R(D) as Prfcer domain

Abstract

$\DeclareMathOperator{\IntR}{Int{}^\text{R}}$Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain $D$, the collection of all integer-valued rational functions over $D$ forms a ring extension $\IntR(D)$ of $D$. For a valuation domain $V$, we characterize when $\IntR(V)$ is a Pr\"ufer domain and when $\IntR(V)$ is a B\'ezout domain. We also extend the classification of when $\IntR(D)$ is a Pr\"ufer domain.