= Surjectivity from injectivity modulo the Jacobson radical
{title2=$\alpha^{-1}(VJ)\subseteq VJ\implies\alpha(V)=V$}
For a <finitely generated module> $V$ on the right over a <semilocal ring>, the displayed condition makes the induced <module endomorphism> on $V/VJ$ <injective>. This quotient is an <Artinian module>, so <Artinian modules are co-Hopfian> makes that map <surjective>. The remaining <quotient module> $V/\alpha(V)$ equals its product with $J$, and <Nakayama lemma> makes it zero.
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