If $R$ is an Euclidean domain that is not a field, is $R[X]$ a PID?

Question

I have a question in ring theory whose answer I am looking for.

Consider R to be a Euclidean Domain such that R is not a field. Then is polynomial ring R[X] is always a PID or not.

Attempt : R is ED implies R[X] is ED and R is ED implies R is PID. But can someone please tell what can I say about R[X] from it.

Answer 1

In this case $R[X]$ is never a PID. Let $\pi$ be a prime element in $R$ (a nonzero generator of a maximal ideal). Then in $R[X]$ the ideal $(\pi,X)$ is non-principal ($R[X]$ is a UFD and $\pi$ and $X$ have no nontrivial common factor).