The Noetherian property in rings of integer-valued polynomials

Abstract

Let D D be a Noetherian domain, D ′ D\prime its integral closure, and Int ⁡ ( D ) \operatorname {Int}(D) its ring of integer-valued polynomials in a single variable. It is shown that, if D ′ D\prime has a maximal ideal M ′ M\prime of height one for which D ′ / M ′ D\prime /M\prime is a finite field, then Int ⁡ ( D ) \operatorname {Int}(D) is not Noetherian; indeed, if M ′ M\prime is the only maximal ideal of D ′ D\prime lying over M ′ ∩ D M\prime \cap D , then not even Spec ⁡ ( Int ⁡ ( D ) ) \operatorname {Spec}(\operatorname {Int}(D)) is Noetherian. On the other hand, if every height-one maximal ideal of D ′ D\prime has infinite residue field, then a sufficient condition for Int ⁡ ( D ) \operatorname {Int}(D) to be Noetherian is that the global transform of D D is a finitely generated D D -module.