Noetherian modules are Hopfian
= Noetherian modules are Hopfian
{c}
{title2=$f:M\twoheadrightarrow M\implies f\text{ is invertible}$}
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>.