Non-triviality Conditions for Integer-valued Polynomial Rings on Algebras

Abstract

Let $D$ be a commutative domain with field of fractions $K$ and let $A$ be a torsion-free $D$-algebra such that $A\cap K=D$. The ring of integer-valued polynomials on $A$ with coefficients in $K$ is ${\mathrm{Int}}_K(A)=\{f\in K[X]\mid f(A)\subseteq A\}$, which generalizes the classic ring $\mathrm{Int}(D)=\{f\in K[X]\mid f(D)\subseteq D\}$ of integer-valued polynomials on $D$. The condition on $A\cap K$ implies that $D[X]\subseteq {\mathrm{Int}}_K(A)\subseteq \mathrm{Int}(D)$, and we say that ${\mathrm{Int}}_K(A)$ is nontrivial if ${\mathrm{Int}}_K(A)\ne D[X]$. For any integral domain $D$, we prove that if $A$ is finitely generated as a $D$-module, then ${\mathrm{Int}}_K(A)$ is nontrivial if and only if $\mathrm{Int}(D)$ is nontrivial. When $A$ is not necessarily finitely generated but $D$ is Dedekind, we provide necessary and sufficient conditions for ${\mathrm{Int}}_K(A)$ to be nontrivial. These conditions also allow us to prove that, for $D$ Dedekind, the domain ${\mathrm{Int}}_K(A)$ has Krull dimension 2.

to appear in Monatsh. Math. (2016). Comments are welcome!

References in corpus (4)

• Properly Integral Polynomials over the Ring of Integer-valued Polynomials on a Matrix Ring
• Integral closure of rings of integer-valued polynomials on algebras
• The ring of polynomials integral-valued over a finite set of integral elements
• Polynomial overrings of $\mathrm{Int}(\mathbb{Z})$

Cited by in corpus (4)

• Decomposition of integer-valued polynomial algebras
• Polynomial Dedekind domains with finite residue fields of prime characteristic
• Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains
• Nontriviality of rings of integral-valued polynomials