Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 101 / 5 / a

A Dedekind domain is a Noetherian ring $D$ which is an integral domain, is an integrally closed domain, and has Krull dimension one. Explicitly, every ideal is finitely generated; every element of the fraction field that is an integral element over $D$ already belongs to $D$; and every nonzero prime ideal is a maximal ideal, with $D$ itself not a field.

With this convention, an equivalent local characterization is that $D$ is a Noetherian integral domain that is not a field and whose localization at a prime ideal $D_P$ is a discrete valuation ring for every nonzero prime ideal $P$. The integrally closed domain condition rules out singularities such as those represented by the subring $k[t^2,t^3]\subset k[t]$, where $t$ is integral but missing.

Some conventions also include fields by allowing Krull dimension at most one; the conclusions about ideals below remain valid under that convention.