A Dedekind ring is a UFD if and only if it is a PID, but not all UFDs are PIDs, so it must be the case that there are UFDs that are not Dedekind. But on the other hand, since unique factorization of ideals into prime ideals seems like a much weaker condition than unique factorization of elements into primes, shouldn't all UFDs be Dedekind and therefore PIDs?
What are the errors in my logic, and (if they do exist), what's an example of a UFD that is not Dedekind?
Best Answer
Note that $\mathbb{Q}[x,y]$ is a UFD, but it is not Dedekind domain since $(x)$ is a non-zero prime ideal, which is not maximal.