The prime and maximal ideals in \(R[X]\), \(R\): a one-dimensional Prüfer domain (Q1286539)

A Prüfer domain \(R\) of Krull dimension 1 is an integral domain in which every non-zero prime ideal \(P\) is maximal and the localization \(R_P\) of \(R\) at \(P\) is a valuation ring. In this note the authors classify all the maximal ideals and prime ideals of the polynomial ring in one variable over a Prüfer domain of Krull dimension 1. They use elementary results from commutative ring theory and valuation theory.