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 me give you a proof of  the following astonishing result due to Vasconcelos:  
Theorem:
Let $M$ be a finitely generated $R$-module, Noetherian or not, and let $ \  f : M \rightarrow M \  $  be a surjective homomorphism. Then $f : M \rightarrow M $ is injective (hence is an  isomorphism).   
Proof:
We use the standard trick of converting $M$ into an $R[X]$-module by defining $X\cdot m=f(m)$.
For the ideal $I=XR[X]$  we have $M=IM$ since for any $m\in M$ we can write by surjectivity of $f$ :  $m=f(n)=X\cdot n$ and $X\in I$.
Since Nakayama says that $$M=IM\implies m=im$$ there exists $i=P(X)X\in I$
with $m=P(X)X\cdot m=P(f)(f(m))$ for all $m\in M$.
So that finally $f(m)=0\implies m=P(f)(f(m))=P(f)(0)=0 \:$: injectivity of $f$ has been proved. 
