Prime ideals in two-dimensional domains over the integers
This paper investigates the question ``For which Noetherian domains \(R\) is \(\text{Spec}(R)\) order-isomorphic to \(\text{Spec} (\mathbb{Z}[X])\)?'' \textit{R. Wiegand} [J. Pure Appl. Algebra 40, 209-214 (1986; Zbl 0592.13002)] has conjectured that every two-dimensional domain \(A\) which is a finitely generated \(\mathbb{Z}\)-algebra has prime spectrum order-isomorphic to \(\text{Spec} (\mathbb{Z}[X])\). The main result of the paper under review is that if \(B=\mathbb{Z} [g_1/f, \dots, g_n/f]\) where the \(g_i\)'s and \(f\) are polynomials in \(\mathbb{Z}[X]\) with \(f\neq 0\), then \(\text{Spec}(B)\) is order-isomorphic to \(\text{Spec} (\mathbb{Z}[X])\). The proof proceeds by showing that \(\text{Spec}(B)\) satisfies the five CZP Axioms 1.3 given by R. Wiegand characterizing \(\text{Spec} (\mathbb{Z}[X])\) as a poset.
scientific article; zbMATH DE number 1308217