Do these properties characterize number rings?

Question

Suppose $R$ is a Dedekind domain with the following properties:

• at every prime of $R$ the residue field is finite;
• fibers of the map $\text{Spec }R\to \text{Spec } \mathbb Z$ are finite.

Is $R$ isomorphic to [a subring of] the number ring of some number field?

Answer 1

The answer is no. E.g. take $R = \mathbb Z[1/2]$. A different kind of counterexample is given by taking $R = \mathbb Z_p$.

You need an additional hypotheses. Taking $R$ to be of finite type over $\mathbb Z$ will eliminate the second kind of counterexample, though not the first. If you take $R$ to be finite over $\mathbb Z$, you do get a subring of a ring of integers.