Prime Ideals in Polynomial Rings Over One-Dimensional Domains

Abstract

Let $R$ be a one-dimensional integral domain with only finitely many maximal ideals and let $x$ be an indeterminate over $R$. We study the prime spectrum of the polynomial ring $R[x]$ as a partially ordered set. In the case where $R$ is countable we classify $\operatorname {Spec} (R[x])$ in terms of splitting properties of the maximal ideals ${\mathbf {m}}$ of $R$ and the valuative dimension of ${R_{\mathbf {m}}}_{}$.