Proving that surjective endomorphisms of Noetherian modules are isomorphisms and a semi-simple and noetherian module is artinian. I am revising for my Rings and Modules exam and am stuck on the following two questions:
$1.$ Let $M$ be a noetherian module and $ \  f : M \rightarrow M \  $  a surjective homomorphism. Show that $f : M \rightarrow M $ is an isomorphism.
$2$. Show that if a semi-simple module is noetherian then it is artinian.
Both these questions seem like they should be fairly straightforward to prove but I cannot seem to solve them.

For the first question, look at the following chain of ideals:
$$\ker f\subset \ker f^2\subset \ker f^3\ldots$$
