Corollary 2 (Burnside’s theorem)

Let A be an algebra over an algebraically closed commutative field K, and let S be a simple A-module that is finite-dimensional as a K-vector space. Then the mapping $a\mapsto a_S$ from A to End$_K(S)$ is surjective.

Indeed, the field of endomorphisms of the A-module S consists of the homotheties $\alpha_S$ with $\alpha \in K$ (VIII, p. 47, Theorem 1). We then apply Corollary 1 to the simple A-module S.