Dedekind rings which are UFDs but not PIDs?

Question

I just have a really quick question of an example that I was trying to come up with.

Are there any number rings which are UFDs but not PIDs?

Answer 1

Since number rings are Dedekind domains we need only show

Dedekind UFDs are PIDs.

If $\mathcal{O}$ is Dedekind and a UFD, then let $x\in\mathfrak{p}$, a prime ideal of $\mathcal{O}$. Then $x=p_1^{e_1}\ldots p_k^{e_k}$ as a product of irreducibles so one of the $p_i\in\mathfrak{p}$. Then note $\mathfrak{p}\mid(p_i)$, but since $p_i$ is irreducible, it is prime and since $\mathcal{O}$ has Krull dimension $1$, it follows that $\mathfrak{p}=(p_i)$.