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Ring Structure of Integer-Valued Rational Functions

Baian Liu

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 Prüfer domain and when $\IntR(V)$ is a Bézout domain. We also extend the classification of when $\IntR(D)$ is a Prüfer domain.

Ring Structure of Integer-Valued Rational Functions

Abstract

Integer-valued rational functions are a natural generalization of integer-valued polynomials. Given a domain , the collection of all integer-valued rational functions over forms a ring extension of . For a valuation domain , we characterize when is a Prüfer domain and when is a Bézout domain. We also extend the classification of when is a Prüfer domain.
Paper Structure (2 sections, 9 theorems, 11 equations)

This paper contains 2 sections, 9 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Integer-valued rational functions over valuation domains

Key Result

Proposition 1.4

Let $D$ be a domain that is not a field and $K$ its field of fractions. If $\varphi \in K(x)$ is such that $\varphi(r) \in D$ for almost all $r \in K$, then $\varphi \in \mathop{\mathrm{Int{}^\text{R}}}\nolimits(K, D)$.

Theorems & Definitions (37)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Proposition 1.4
  • proof
  • Definition 1.5
  • Remark 1.6
  • Proposition 1.7
  • proof
  • Corollary 1.8
  • ...and 27 more