Schur's lemma

Remarks

The kernel of $f$ is a submodule of $M$, so it is $0$ or $M$. The image of $f$ is a submodule of $N$, so it is $0$ or $N$. If $f \ne 0$, the kernel is not $M$ and the image is not $0$, so $f$ is injective and surjective. Example: a nonzero group homomorphism $\Z/p\Z \to \Z/p\Z$, for $p$ prime, is a bijection. Simplicity is needed: the map $\Z/4\Z \to \Z/4\Z$, $x \mapsto 2x$, is nonzero and not an isomorphism. A common form concerns an algebraically closed field $k$, a $k$-algebra $A$, and a simple $A$-module $M$ of finite dimension over $k$. Then every endomorphism of $M$ is multiplication by a scalar.

Going deeper

For a student

Schur's lemma is the first tool of representation theory. It shows that the endomorphism ring of a simple module is a division ring. For representations of a group over $\C$ it gives that the endomorphisms of an irreducible representation are scalars. This is the key step in the orthogonality relations for characters and in the decomposition of semisimple modules into isotypic parts.