# Suppose A is a principal ideal domain with every ideal of finite index. Must A be a Euclidean domain?

Suppose $A$ is a principal ideal domain with every ideal of finite index (except the zero ideal). Must $A$ be a Euclidean domain?

If it's not known, are there any relevant partial results?

Not necessarily. The classical example of PID which is not Euclidean, $R=\mathbb Z[\frac{1+\sqrt{-19}}{2}]$ has the property that $R/I$ is finite for all $I\ne(0)$. (In order to prove this use a similar argument to the one given in this answer.)