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.

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