Given an integral domain D with quotient field K, the ring of integer-valued polynomials on D is the subringUsing the tools of t-closure and associated primes, we generalize some known results on integer-valued polynomial rings over Krull domains, PVMD's, and Mori domains.

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Let D be an integral domain, and let A be a domain containing D with quotient field K. We will say thatWe show that, for any set X, the ring Int(D X ) of integervalued polynomials on D X is the free polynomially complete extension of D generated by X, 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(A n ) ⊇ Int(D n ) 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 ϕ n : n i=1 Int(D) −→ Int(D n ) is surjective for all positive integers n, where the tensor product is over D. Furthermore, ϕ n 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(D m ) = Int(D) m for every maximal ideal m of D.

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