Substitution and Closure of Sets under Integer-Valued Polynomials

Let R be a domain and K its quotient-field. For a subset S of K , let FR(S) be the set of polynomials f ∈ K[x] with f(S) ⊆ R and define the R -closure of S as the set of those t ∈ K for which f(t) ∈ R for all f ∈ FR(S). The concept of R -closure was introduced by McQuillan (J. Number Theory 39 (1991), 245–250), who gave a description in terms of closure in P -adic topology, when R is a Dedekind ring with finite residue fields. We introduce a toplogy related to, but weaker than P -adic topology, which allows us to treat ideals of infinite index, and derive a characterization of R -closure when R is a Krull ring. This gives us a criterion for FR(S) = FR(T ), where S and T are subsets of K . As a corollary we get a generalization to Krull rings of R. Gilmer’s result (J. Number Theory 33 (1989), 95–100) characterizing those subsets S of a Dedekind ring with finite residue fields for which FR(S) = FR(R).