Aside from $\langle 0 \rangle$, can a ring of algebraic integers have prime ideals that are not maximal?

Question

I have a feeling that a ring with such ideals would have to be non-UFD, and I can prove that in $\mathbb{Z}$ there are no such ideals. But in other rings, I'm not so sure. I'm not yet at a point at which I understand ideals in rings of polynomials, though I have seen a couple of examples of non-maximal prime ideals in such rings.

Answer 1

No. If $\mathcal{O}_K$ is a ring of integers and $P$ is a nonzero prime ideal, then $\mathcal{O}_K/P$ is a finite integral domain, and by Wedderburn's little theorem it must be a field. More generally, Dedekind domains have Krull dimension $1$, which means that if $D$ is a Dedekind domain and $P$ is a nonzero prime ideal then $D/P$ has no nontrivial prime ideals, hence is a field.