Let be a ring with 1. Prove that if and are irreducible unital left -modules, then any nonzero -module homomorphism is an isomorphism. Deduce Schur’s Lemma: If is an irreducible unital left -module, then is a division ring.
Suppose is an -module homomorphism with and irreducible. We showed here that and are submodules of and , respectively. Since is not the zero homomorphism, its kernel is not all of , and so (since is irreducible) . Thus is injective. Similarly, is not the zero submodule, and so must be , hence is surjective. Thus is an -module isomorphism.
Now let be an irreducible unital left -module. Recall that is a ring under pointwise addition and composition. Now if is nonzero, then by the above argument it is an isomorphism and so has an inverse which is also in . So is a division ring. Note that is not necessarily commutative, so it need not be a field.