Two Prüfer domain counterexamples
In this paper examples are constructed to show (1) a Prüfer domain \(D\) whose maximal ideals are principal but \(D\) itself is not a Bezout domain, (2) a three-dimensional Prüfer domain \(D\) which is the intersection of a collection \(V_i\) of valuation subrings of \(\mathbb{Q}(x,y)\), the field of rational functions in two variables over \(\mathbb{Q}\), with \(V_1\) a three-dimensional, and \(V_i\), \(i>1\), one-dimensional valuation rings. Denoting by \(M_x\) and \(M_{(x,y)}\) the height 1 and height 2 prime ideal contained in the contraction \(M_1\) of \(D\) in \(V_1\), the author shows that \(M_{(x,y)}\) is divisorial while \(M_x\) is not.
scientific article; zbMATH DE number 1400112