Schur's Lemma

A result in representation theory and module theory: any homomorphism between simple modules is either zero or an isomorphism; equivalently, the endomorphism ring of a simple module is a division algebra, and over algebraically closed fields this endomorphism algebra is just the field (scalars).

A statement about morphisms between simple (irreducible) objects in module or representation categories: any homomorphism between two simple modules is either zero or an isomorphism; the endomorphism ring of a simple module is a division ring (a skew field).

A statement about homomorphisms of simple (irreducible) modules or representations: any nonzero homomorphism between simple modules is an isomorphism; consequently, the endomorphism ring of a simple module is a division ring (and over an algebraically closed field it is just the field of scalars).