Factorization of rings of integer-valued rational functions

TL;DR

This paper investigates the factorization properties of rings of integer-valued rational functions over domains, introducing new atomic rings and analyzing their factorization characteristics.

Contribution

It introduces a family of atomic rings of integer-valued rational functions and studies their factorization properties, expanding understanding beyond known cases.

Findings

Identified conditions under which rings of integer-valued rational functions are atomic.

Constructed examples of atomic rings of integer-valued rational functions.

Analyzed various factorization properties of these rings.

Abstract

$\DeclareMathOperator{\Int}{Int}\DeclareMathOperator{\IntR}{Int{}^\text{R}}$For a domain $D$, the ring $\Int(D)$ of integer-valued polynomials over $D$ is atomic if $D$ satisfies the ascending chain condition on principal ideals. However, even for a discrete valuation domain $V$, the ring $\IntR(V)$ of integer-valued rational functions over $V$ is antimatter. We introduce a family of atomic rings of integer-valued rational functions and study various factorization properties on these rings.