Does a UFD imply Euclidean Domain?

Question

I'm struggling to get my head around the relationship between UFD, PID and Euclidean Domain. I've seen in a theorem in my notes that Euclidean Domain ⇒ PID ⇒ UFD ⇒ ID. Is it possible for UFD ⇒ ED?

Answer 1

$\mathbb{R}[x,y]$ is UFD, but not a PID and hence also not ED. In general there are PIDs that are not ED for instance this wikipedia article claims that $\mathbb{R}[x,y]/(x^2+y^2-1)$ is not ED although it is PID. There are also examples in the class of number rings.