i'm trying to prove the claim in the title, my argument is:
$k$ is a field so it's Noetherian, hence so is $k[x,y]$ by the Hilbert basis theorem, hence $k[x,y]$ is a Noetherian $k$-module, and a quotient of it is also finitely generated, as every submodule of a Noetherian module is finitely generated.
However it appears to me that, in the same argument, $k[x]$ and $k[x,y]$ are finitely generated $k$-modules as well because they are submodules of $k[x,y]$, which is clearly absurd. Could you tell me in which step did I make a mistake? Thanks!