Polynomial overrings of \(\mathrm{Int}(\mathbb Z)\) (Q262655)

The authors present a description of rings \(R\subset Q[X]\) containing the ring Int\((Z)\) of integer-valued polynomials as intersections of valuation overrings of Int\((Z)\) (Proposition 5.1). They show also that every such ring is the ring of integer-valued polynomials over a subset of \(\hat Z\), the profinite completion of \(Z\), the non-zero ideals of \(Z\) forming a fundamental set of neighbourhoods of \(0\) (Theorem 6.2). The proofs are based on a description of polynomial overrings of the localization NEWLINE\[NEWLINE\text{Int}\;Z_{(p)}=\left\{{f(X)\over n}:\;f\in\;\text{Int}\;Z, n\in Z, n\not\equiv0\pmod p\right\}NEWLINE\]NEWLINE for \(p\) prime (Proposition 3.3).