The following claim is from a paper of Mukai. Consider an artinian module $\mathrm{M}$ over a local ring, and let $\mathrm{M}_0$ be the submodule of all $x\in\mathrm{M}$ annhilated by the maximal ideal of the local ring. Then every endomorphism of $\mathrm{M}$ preserves $\mathrm{M}_0$, and the natural map $$\mathrm{Hom}(\mathrm{M},\mathrm{M})\rightarrow\mathrm{Hom}(\mathrm{M}/\mathrm{M}_0,\mathrm{M}/\mathrm{M}_0)$$ is surjective. I can't quite see why this map is surjective - this is not justified in the paper, so I may missing something obvious here (a comment may be enough).

The following claim is from a paper [On the moduli spaces of bundles on K3 surfaces, I, p. 358] of Mukai. Consider an artinian module $\mathrm{M}$ over a local ring, and let $\mathrm{M}_0$ be the submodule of all $x\in\mathrm{M}$ annhilated by the maximal ideal of the local ring. Then every endomorphism of $\mathrm{M}$ preserves $\mathrm{M}_0$, and the natural map $$\mathrm{Hom}(\mathrm{M},\mathrm{M})\rightarrow\mathrm{Hom}(\mathrm{M}/\mathrm{M}_0,\mathrm{M}/\mathrm{M}_0)$$ is surjective. I can't quite see why this map is surjective - this is not justified in the paper, so I may be missing something obvious here (a comment may be enough).