Polynomial overrings of $\mathrm{Int}(\mathbb{Z})$

Abstract

We show that every polynomial overring of the ring $\mathrm{Int}(\mathbb{Z})$ of polynomials which are integer-valued over $\mathbb{Z}$ may be considered as the ring of polynomials which are integer-valued over some subset of $\hat{\mathbb{Z}}$, the profinite completion of $\mathbb{Z}$ with respect to the fundamental system of neighbourhoods of $0$ consisting of all non-zero ideals of $\mathbb{Z}.$

Final version, J. Commut. Algebra 8 (2016), no. 1, 1-28. Keywords: Integer-valued polynomial, Prüfer domain, Overring, Irredundant representation

References in corpus (2)

• Integral closure of rings of integer-valued polynomials on algebras
• The ring of polynomials integral-valued over a finite set of integral elements

Cited by in corpus (7)

• Properly Integral Polynomials over the Ring of Integer-valued Polynomials on a Matrix Ring
• Non-triviality Conditions for Integer-valued Polynomial Rings on Algebras
• Adelic versions of the Weierstrass approximation theorem
• Decomposition of integer-valued polynomial algebras
• Galois structure on integral valued polynomials
• Polynomial Dedekind domains with finite residue fields of prime characteristic
• Stacked Pseudo-Convergent Sequences and Polynomial Dedekind Domains