First we show to be a Noetherian ring. Let be an increasing sequence of ideals, and without loss of generality take . By The ring of integers of a number field forms a lattice, it follows is finite, implying there are only finitely many subrings of containing and thus the sequence must stabilize. Therefore is Noetherian.

Now let be a nonzero prime ideal. It follows from ^C1 that is finite, and A finite integral domain is a field, thus is maximal since for commutative is a field iff is maximal. Therefore, has Krull dimension .

Since the ring of integers is automatically integrally closed, it follows is Dedekind.