Module over a polynomial ring

Question

I'm trying to prove the following:

If $D = K[x]$ where $K$ is a field, and $M \subset D^n$ is a submodule for some $n \geq 2$ then $L = D^n/M$ is Artinian iff the torsion module of $L$ is isomorphic to $M$.

So far, since $K$ is a field we have that $D$ is a PID and then any module $L$ over D is of the form $D^k \times T(L)$, but I don't see how I can proceed from here. Any help will be welcome.

Thank You!

Answer 1

Hint. Prove the following:

$L$ is artinian $\iff$ $L$ is torsion $\iff$ $\operatorname{rank}M=n$.