THE NOETHERIAN PROPERTY IN RINGS OF INTEGER-VALUED POLYNOMIALS

Let D be a Noetherian domain, D' its integral closure, and Int(D) its ring of integer-valued polynomials in a single variable. It is shown t hat, if D' has a maximal ideal M' of height one for which D'/M' is a f inite field, then Int(D) is not Noetherian; indeed, if M' is the only maximal ideal of D' lying over M' and D, then not even Spec(Int(D)) is Noetherian. On the other hand, if every height-one maximal ideal of D ' has infinite residue field, then a sufficient condition for Int(D) t o be Noetherian is that the global transform of D is a finitely genera ted D-module.