I am trying to prove that:
Every finitely generated $F$-module $M$ is both Noetherian and Artinian where $F$ is a field.
For this I am looking at the submodules of $F$ and saying that they are in correspondence with the ideals in $F$. Since $F$ has only two ideals it clearly satisfies the condition to be both Noetherian and Artinian. Is this argument correct? I don't see why we need the finitely generated assumption?