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 $F[x]/(x^3)$.

  • $\begingroup$ I think that finitely generated modules over Noetherian (Artinian) rings are again Noetherian (Artinian). $\endgroup$ – Rene Schipperus Feb 23 '17 at 1:13
  • $\begingroup$ 1) Let $S=F[y]$. Then $R=S[x]/(x^3)$ is a finitely generated $S$-module. 2) Let $T=F[x]/(x^3)$. Then $R=T[y]$ is not a finitely generated $T$-module. $\endgroup$ – user26857 Feb 23 '17 at 8:43

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.


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