Counterexample for a principal ring

Question

I know: If $R$ is a principal ring, then every nonzero prime ideal is maximal.

What is a counterexample for the converse (opposite direction)?

Answer 1

$\mathbb{Z}[\sqrt{-5}]$ is not a PID because it doesn't have unique factorization. For instance, $2\cdot 3=6=(1+\sqrt{-5})(1-\sqrt{-5})$, and one can show that $1+\sqrt{-5}$ is not an associate of either $2$ or $3$.

However, $\mathbb{Z}[\sqrt{-5}]$ is a Dedekind domain because it is the ring of integers of $\mathbb{Q}(\sqrt{-5})$. Hence every non-zero prime ideal in $\mathbb{Z}[\sqrt{-5}]$ is maximal.