Universal properties of integer-valued polynomial rings

Let D be an integral domain, and let A be a domain containing D with quotient field K. We will say that the extension A of D is polynomially complete if D is a polynomially dense subset of A, that is, if for all f∈K[X] with f(D)⊆A one has f(A)⊆A. We show that, for any set , the ring of integer-valued polynomials on is the free polynomially complete extension of D generated by , provided only that D is not a finite field. We prove that a divisorial extension of a Krull domain D is polynomially complete if and only if it is unramified, and has trivial residue field extensions, at the height one primes in D with finite residue field. We also examine, for any extension A of a domain D, the following three conditions: (a) A is a polynomially complete extension of D; (b) Int(An)⊇Int(Dn) for every positive integer n; and (c) Int(A)⊇Int(D). In general one has (a) ⇒ (b) ⇒ (c). It is known that (a) ⇔ (c) if D is a Dedekind domain. We prove various generalizations of this result, such as: (a) ⇔ (c) if D is a Krull domain and A is a divisorial extension of D. Generally one has (b) ⇔ (c) if the canonical D-algebra homomorphism is surjective for all positive integers n, where the tensor product is over D. Furthermore, φn is an isomorphism for all n if D is a Krull domain such that Int(D) is flat as a D-module, or if D is a Prüfer domain such that Int(Dm)=Int(D)m for every maximal ideal m of D.