Surjectivity from injectivity modulo the Jacobson radical (${\alpha }^{-1}(VJ)\subseteq VJ⟹\alpha (V)=V$)

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.