Dedekind domain

A Dedekind domain is a commutative integral domain $R$ for which:

• Every ideal in $R$ is finitely generated.
• Every nonzero prime ideal is a maximal ideal.
• The domain $R$ is integrally closed in its field of fractions.

It is worth noting that the second clause above implies that the maximal length of a strictly increasing chain of prime ideals is 1, so the Krull dimension of any Dedekind domain is at most 1. In particular, the affine ring of an algebraic set is a Dedekind domain if and only if the set is normal, irreducible, and 1-dimensional.

Every Dedekind domain is a noetherian ring.

If $K$ is a number field, then $\mathcal{O}_K$ , the ring of algebraic integers of $K$ , is a Dedekind domain.