Example of a Noetherian UFD that is not a Dedekind domain.

Question

I am having trouble finding one example of a Noetherian UFD that is not a Dedekind domain. Or maybe a Noetherian UFD is always a Dedekind domain and henceforth a PID.

Answer 1

The DaRT query has an example: $\mathbb Z[x]$.

You could have come upon this yourself reading the wiki article, where it states that Dedekind Domains have dimension less than $2$, whereas the example has dimension $2$.