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Ideals of integer-valued polynomial rings. (English) Zbl 0896.13015

Let \(\text{Int}(\mathbb{Z}) = \{f \in \mathbb{Q} [x] \mid f (\mathbb{Z}) \subseteq \mathbb{Z}\}\). The finitely generated ideals of this ring have been extensively studied using as principal tool the set of value ideals of an ideal \(\{I(a)=\{f(a)| f\in I\}\) for \(a\in \mathbb{Z}\}\) and the fact that \(\text{Int}(\mathbb{Z})\) has the strong Skolem property that if \(I\) and \(J\) are finitely generated ideals of \(\text{Int}(\mathbb{Z})\) for which \(I(a)=J(a)\) for all \(a\in \mathbb{Z}\) then \(I=J\). \(\text{Int}(\mathbb{Z})\) is non-noetherian, however, and so contains non-finitely generated ideals. The present paper gives some extensions of the strong Skolem property by considering value ideals \(I(a)\) with \(a\in \widehat\mathbb{Z}_p\), the p-adic integers. These are applied to extend a result of R. Gilmer and W. W. Smith [J. Algebra 81, 150-164 (1983; Zbl 0515.13016)] characterizing the value ideals of finitely generated ideals to ideals \(I\) for which \(I\cap \mathbb{Z}=m\mathbb{Z}\) with \(m\) square free and to describe an explicit construction of a non-finitely generated divisorial ideal in \(\text{Int}(\mathbb{Z})\).

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13A15 Ideals and multiplicative ideal theory in commutative rings
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors

Citations:

Zbl 0515.13016
Full Text: DOI

References:

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