Let be a commutative ring and let be a nilpotent ideal, say with . Let and be left -modules and suppose is an -module homomorphism. Prove that if the induced mapping is surjective, then is surjective.
[I am ashamed to admit that I had to refer to this forum post to see how to prove this one.]
The induced mapping is given by .
Note that since is surjective, we have . By the Lattice Isomorphism theorem for modules, we have .
We claim that for all . We prove this by induction, with the base case already shown. Suppose now that the equation holds for some ; then as desired, since .
Since , we have . Thus is surjective.