Note on integer-valued polynomials on a residually cofinite subset

Abstract

Let D be an integral domain with quotient field K, E a subset of K and X an indeterminate over K. The ring $\mathrm {Int}(E,D):=\{f\in K[X]\;:\; f(E)\subseteq D\}$, of integer-valued polynomials on E with respect to D, is known to be a D-algebra. Obviously, $\mathrm {Int}(D,D)=\mathrm {Int}(D)$, is the classical ring of integer-valued polynomials over D. In this paper, we study the faithful flatness, the local freeness and the calculation of the Krull dimension of integer-valued polynomials on a residually cofinite subset over locally essential domains. In particular, we prove that $\mathrm {Int}(E,D)$ is faithfully flat as a D-module and is of Krull dimension less than that of D[X], when D is a locally essential domain and $E\subseteq D$ is residually cofinite with D. Also, we get stronger results in the case of domains that are either almost Krull or t-almost Dedekind.