Let $K$ be a field and consider $K[[x]]$ as a $K[x]$-module. Determine if it is Artinian/Noetherian.
I used the following propositions:
If $M$ is an $R$-module and $N\subseteq M$ a submodule, then $M$ is Artinian/Noetherian if and only if both $N$ and $M/N$ are Artinian/Noetherian.
A ring $R$ is Artinian if and only if $R$ is Noetherian and every prime ideal of $R$ is maximal.
First the ring $K[x]$. All its ideals are finitely generated, so it's Noetherian. But the ideal $(0)$ is prime, but not maximal. So $K[x]$ is not Artinian.
I had then planned to use that $K[x]/(x^n)\cong K[[x]]/(x^n)$. Using this relation we can see that either $K[[x]]/(x^n)$ or $(x^n)$ isn't Artinian, since $K[x]$ is not either. This means $K[[x]]$ is not Artinian.
Using the same relation we can see that both $K[[x]]/(x^n)$ and $(x^n)$ are Noetherian since $K[x]$ is. This would mean $K[[x]]$ is Noetherian too.
Could anyone verify this proof.