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

TL;DR

This paper characterizes polynomial overrings of the ring of integer-valued polynomials over integers as rings of polynomials integer-valued over subsets of the profinite completion of integers, connecting algebraic and topological structures.

Contribution

It provides a complete description of polynomial overrings of Int(ℤ) in terms of integer-valued polynomials over subsets of the profinite completion of ℤ.

Findings

Polynomial overrings correspond to integer-valued polynomials over subsets of .

The structure of these overrings is linked to the topology of .

The results unify algebraic and topological perspectives on polynomial rings.

Abstract

We show that every polynomial overring of the ring ${\rm Int}(\mathbb Z)$ 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 $\mathbb Z.$