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.

$1.$ Let $\,f:M\to M\,$ be an epimorphism or $\,R\,$-modules, with $\,M\,$ Noetherian.
i) Show that $\,M\,$ can be made into $\,R[t]\,$-module, defining $\,tm:=f(m)\,,\,\forall m\in M$
ii) Putting $\,I:=\langle t\rangle=tR[t]\,$ , show that $\,MI=M\,$
iii) Apply Nakayama's Lemma to deduce that there exists $\,1+g(t)t\in I\,$ s.t. $\,(1+g(t)t)M=0$
iv) Finally, take $\,y\in \ker f\,$ and show $\,y=0\,$ applying (iii)
