I spent like 6 hours trying to prove that the endomorphism ring of a simple module is a field, helping my friend with his homework. Now I'm convinced that the result is not true. By Schur's lemma, all endomorphisms of a simple module (except 0) are isomorphisms, so it's a division ring, but I don't think it will be abelian. Only, so far I haven't found a counterexample. Can someone help me find one?