Remark

Let K be an algebraically closed commutative field and A a K-algebra. Let M be a semisimple A-module that is a finite-dimensional vector space over the field K. Then M is simple if and only if every endomorphism of the A-module M is of the form $x\mapsto \alpha x$ with $\alpha$ in K: this is necessary by Theorem 1 of VIII, p. 47, and sufficient by Corollary 4 above.