Noetherian modules are Hopfian ($f:M\twoheadrightarrow M⟹f\text{ is invertible}$)

For a surjective module endomorphism of a Noetherian module, the chain $\ker f\subseteq \ker f^2\subseteq \cdots$ stabilizes at some $N$. If $f(x)=0$, choose $y$ with $f^N(y)=x$. Then $y\in \ker f^{N+1}=\ker f^N$, giving $x=0$. Thus the endomorphism is also injective and is an isomorphism.