I've been learning algebraic number theory I'm wondering if there's a converse to the idea that a ring having a Euclidean algorithm/division with remainder makes it a unique factorization domain.
So I think my question is, do all the number rings with unique factorization have a Euclidean algorithm?
Also curious for examples of general rings (not necessarily number rings) that have unique factorization but don't have a Euclidean algorithm.
Thanks for any insight.