Prime ideals in polynomial rings over one-dimensional domains

Abstract

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