Artinian modules are co-Hopfian
= Artinian modules are co-Hopfian
{c}
{title2=$f\text{ injective}\Longrightarrow f\text{ surjective}$}
For an injective <module endomorphism> $f:M\to M$ of an <Artinian module>, the chain $M\supseteq fM\supseteq f^2M\supseteq\cdots$ stabilizes. If $f^nM=f^{n+1}M$, then every $x$ satisfies $f^nx=f^{n+1}y$. Injectivity of $f^n$ gives $x=fy$, proving surjectivity. No finite-length or Noetherian hypothesis is needed.