Converse Noetherian Relation

Question

I have looked around and cannot find the answer so I ask here. It is well known that if $R$ is noetherian then $R[X]$ is too, but what about the converse? If $R[X]$ is noetherian can we say $R$ is? My gut feeling says no and the fact it seems so hard to find an answer makes me feel certain about it.

Answer 1

The converse is a straightforward consequence of the following fact:

Let $S$ be a Noetherian ring and $I \subset S$ an ideal. Then the factor ring $S/I$ is Noetherian.

Now, take $S=R[X]$ , $I=(X)$, and recall that $R[X]/(X) \cong R$.