§4. EXERCISES 11 You’re in a non-commutative setting, where the sum of two nilpotents might be a unit! If you get stuck, consult [Fac98, Lemma 2.21].) 1.27. Exercise. Let M be a Noetherian left Λ-module, and let f ∈ EndΛ(M). (i) If f is surjective, prove that f is an automorphism of M. (Con- sider the ascending chain of submodules ker(f n ).) (ii) If f is surjective and f 2 = f, prove that f = 1M.

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