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).



Endomorphisms of Artinian modules
This is false. Take $A=k[x,y]/(x^2,y^2,xy)$ and $M=\omega\_A$. Then $\operatorname{Hom}(\omega,\omega)$ has length 3 and $\operatorname{Hom}(M/M\_0,M/M\_0)$ has length 4, so the natural map cannot be surjective.


