A note on Dedekind non-\(D\)-rings
A non-\(D\)-ring is a commutative integral domain \(R\) for which there is a non-constant polynomial \(f\) over \(R\) with the property that \(f(a)\) is a unit of \(R\) for all \(a\) in \(R\). If also \(R\) is Dedekind, it is known that the class group \(\text{Cl} (R)\) of \(R\) is torsion, of exponent dividing the degree of \(f\). In this note, the author gives a method for constructing Dedekind non-\(D\)-rings as intersections of valuation rings. This method is applied to valuations in the rational function field \(\mathbb{Q} (X)\) to give some explicit examples of Dedekind non-\(D\)-rings and their class groups.
scientific article; zbMATH DE number 515197