TITLE: Noetherian module and noetherian ring
QUESTION [5 upvotes]: If $R$ is a noetherian ring then also $R[x]$ is a noetherian ring, i.e. $R[x]$ is noetherian as $R[x]$-module. Is $R[x]$ also noetherian as $R$-module?

REPLY [2 votes]: Let $R$ be a commutative ring and $X$ an indeterminate. Then $R[X]$ is a noetherian $R$-module if and only if $R$ is the zero ring.
The other answers contain a proof (but not a statement) of this fact.
Edit. In the first version, I had written "$R[X]$ is a noetherian $R$-module if and only if $R$ is not the zero ring". Thanks to quid for having pointed out this mistake!