Just trying to get my head around Noetherian and Artinian modules, I've come across this question, which I don't really know how to approach:
Let $R=F[x,y]/(x^3)$ where $F$ is a field. Is R Noetherian/Artinian as any of the following: an $F[y]$-module or an $F[x]/(x^3)$-module.
I think the bit that's confusing is the polynomial modules. My initial thoughts are that it's neither noetherian or artinian for $F[y]$, since there are submodules that aren't finitely generated, and both for $F[x]/(x^3)$.
It is noetherian as an $F[y]$-module since it is isomorphic to $$F[y][x]/x^3,$$ so that it is a finitely generated $F[y]$-module.