Schur’s Lemma is a useful theorem in algebra that is surprisingly easy to prove.
Schur’s Lemma: Let and be right -modules and let be a nonzero -module homomorphism.
(i) If is simple, then is injective.
(ii) If is simple, then is surjective.
(iii) If both and are simple then is an isomorphism.
(iv) If is a simple module, then is a division -algebra.
(i) is a submodule of . Since is nonzero, , which means .
(ii) is a submodule of . Since , .
(iii) Combine (i) and (ii) to get a bijective homomorphism.
(iv) is an -algebra. Let be an element in . Since is an isomorphism, its inverse exists. Then . Thus is an division -algebra.