Suppose $R=F$ is a field. Prove that an $R-$module $M$ is Artinian iff it's Noetherian iff $M$ is a finite dimensional vector space over $F$.

Question

Suppose $R=F$ is a field. Prove that an $R-$module $M$ is Artinian iff it's Noetherian iff $M$ is a finite dimensional vector space over $F$.

If $M$ is a finite vector space over $F$, then neither do every subspace of $M$ which coincide with its submodules, so $M$ is Noetherian. I wanna prove that from $M$ is Noetherian, it's also Artinian, and then $M$ is a finite dimensional vector space over $F$ again.

Help me some hints.

Thanks a lot.

Answer 1

given chains of submodules, take a basis for each subspace, then it's easy to see that the ascending (resp. descending) condition will hold iff $M$ is finite dimensional.