Let $R$ be a Euclidean domain, i.e., a ring with a norm $N : R \rightarrow \mathbb N$ such that for any $a,b\in R$ with $b\not=0$, we may write $a = qb + r$ for some $q,r \in R$ with $N(r) < N(b)$. Let $I$ be a prime ideal of $R$, so $R/I$ is a domain. Must $R/I$ in fact be a Euclidean domain?
I know that this is true if we replace "Euclidean domain" with "PID", and false if we replace it with "UFD", so there is no clear intuition to be gained from similar concepts. My first attempts at a proof was to borrow a construction from analysis and define a norm on $R/I$ by
$N(a + I) = \inf\{N(a+i) : i\in I\}$
but this didn't really get me anywhere.
So, this fact true? If so, how can I prove it? If not, whats a counterexample?