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The Noetherian property in rings of integer-valued polynomials

Authors: Robert Gilmer, William Heinzer and David Lantz
Journal: Trans. Amer. Math. Soc. 338 (1993), 187-199
MSC: Primary 13G05; Secondary 13B22, 13B25, 13E05
MathSciNet review: 1097166
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Abstract: Let $ D$ be a Noetherian domain, $ D\prime $ its integral closure, and $ \operatorname{Int}(D)$ its ring of integer-valued polynomials in a single variable. It is shown that, if $ D\prime $ has a maximal ideal $ M\prime $ of height one for which $ D\prime /M\prime $ is a finite field, then $ \operatorname{Int}(D)$ is not Noetherian; indeed, if $ M\prime $ is the only maximal ideal of $ D\prime $ lying over $ M\prime \cap D$, then not even $ \operatorname{Spec}(\operatorname{Int}(D))$ is Noetherian. On the other hand, if every height-one maximal ideal of $ D\prime $ has infinite residue field, then a sufficient condition for $ \operatorname{Int}(D)$ to be Noetherian is that the global transform of $ D$ is a finitely generated $ D$-module.

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Keywords: Rings of integer-valued polynomials, Noetherian domain, Noetherian spectrum, Prüfer domain, Krull domain, integral closure, global transform
Article copyright: © Copyright 1993 American Mathematical Society

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