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 $\mathbb{\widehat Z}$, the profinite completion of $\mathbb Z$ with respect to the fundamental system of neighbourhoods of $0$ consisting of all non-zero ideals of $\mathbb Z$.