On ideals in Prüfer domains of polynomials
type: Item
instance of: scholarly article
title: On ideals in Prüfer domains of polynomials
zbMATH Open document ID: 0596.13018
DOI: 10.1007/BF01194891
author: Donald L. McQuillan
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let R be a Dedekind domain with finite residue fields. If K is the quotient field of R and S is an infinite R-fractional subset of K, the author investigates the ideal theory of the ring \({\mathcal F}(S)=\{f(X)\in K[X]| \quad f(S)\subseteq R\}.\) An ideal I of \({\mathcal F}(S)\) is said to be unitary if \(I\cap R\neq (0)\). Every unitary prime ideal of \({\mathcal F}(S)\) is maximal. The author shows that the unitary maximal ideals of \({\mathcal F}(S)\) are precisely the ideals \(M_{a,p}=\{f(X)\in {\mathcal F}(S)| \quad f(a)\in pR_ p\}.\) Here p is a maximal ideal of R and \(a\in S_ p\). Moreover, \(M_{a,p}=M_{b,p}\) iff \(a=b\), and if \(a\not\in S\), then \(M_{a,p}\) is not finitely generated. The non-unitary nonzero prime ideals of \({\mathcal F}(S)\) are shown to be of the form \(M_ f=f(X)K[X]\cap {\mathcal F}(S)\) where f(X) is an irreducible polynomial in K[X]. Moreover, \(M_ f\) is maximal iff f(X) has no root in \(S_ p\) for any maximal ideal p of R. A generalization is also given to a result of \textit{R. Gilmer} and \textit{W. W. Smith} [J. Algebra 81, 150-164 (1983; Zbl 0515.13016)] concerning finite generation of the ideals \(M_ f\). In \(theorems\quad 3.3\) and 3.6 which are summarized above it is also assumed that for each positive \(integer\quad n\) there is only a finite number of prime ideals p of R such that \(| R/p| =n\). This hypothesis is unnecessary since any Noetherian ring, or even any ring R with Spec R Noetherian, has this property.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 3959606
zbMATH Keywords: Prüfer domain
zbMATH Keywords: ring of ideal valued polynomials
zbMATH Keywords: Dedekind domain
reviewed by: William J. Heinzer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3959606