Noetherian module and noetherian ring 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?

Let $P_n$ be the $R$-sub-module of $R[x]$ whose elements are all polynomials of degree at most $n$.

Then you have the infinite chain
$$P_0 \subset P_1 \subset P_2 \subset \cdots$$
with all inclusions proper.
