# Noetherian module and endomorphism

Given $R$ a ring (commutative and unitary) and $M$ a Noetherian $R$-module, does there exist an endomorphism $f\in End(M)$ which is injective but not surjective?

I already prooved that if $f$ is surjective then it has to be injective (by taking the chain of $Ker(f)\subseteq Ker(f^2) \subseteq ...$ and using the ascending chain condition) but I'm stuck with this part...

Plenty. The endomorphism ring of $R$ as a module over itself is isomorphic to $R$; any nonzero divisor induces an injective endomorphism by multiplication. Is a nonzero divisor necessarily invertible?