Schur’s Lemma

Schur’s Lemma is a useful theorem in algebra that is surprisingly easy to prove.

Schur’s Lemma: Let $M$ and $N$ be right $A$ -modules and let $\phi: M\to N$ be a nonzero $A$ -module homomorphism.

(i) If $M$ is simple, then $\phi$ is injective.

(ii) If $N$ is simple, then $\phi$ is surjective.

(iii) If both $M$ and $N$ are simple then $\phi$ is an isomorphism.

(iv) If $M$ is a simple module, then $\text{End}_A(M)$ is a division $R$ -algebra.

Proof:

(i) $\ker\phi$ is a submodule of $M$ . Since $\phi$ is nonzero, $\ker\phi\neq M$ , which means $\ker\phi=0$ .

(ii) $\text{Im}\,\phi$ is a submodule of $N$ . Since $\text{Im}\,\phi\neq 0$ , $\text{Im}\,\phi=N$ .

(iii) Combine (i) and (ii) to get $\phi$ a bijective homomorphism.

(iv) $\text{End}_A(M):=\text{Hom}_A(M,M)$ is an $R$ -algebra. Let $\phi:M\to M$ be an element in $\text{Hom}_A(M,M)$ . Since $\phi$ is an isomorphism, its inverse $\phi^{-1}$ exists. Then $\phi\circ\phi^{-1}=\text{id}_M=\phi^{-1}\circ\phi$ . Thus $\text{End}_A(M)$ is an division $R$ -algebra.