On ideals in Prüfer domains of polynomials
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.
scientific article; zbMATH DE number 3959606
scientific article