Theorem 1

Let K be an algebraically closed commutative field, A a K-algebra, and M a simple A-module. Suppose that the dimension of M as a vector space over K is finite or, more generally, strictly less than the cardinal of K. Then the endomorphism ring of the A-module M consists of the homotheties $\alpha_M$ with $\alpha \in K$.

Let E be the endomorphism ring of the A-module M; it is a field by the corollary of Proposition 2 and an algebra over the field K. If we view M as a left vector space over the field E, then we have dim$_KM =$ (dim$_EM$)$[E : K]$ by Proposition 25 of II, §1, No. 13, p. 222, hence dim$_KM\geqslant [E : K]$. Since dim$_KM<$ Card(K) by assumption, the equality $E = K\cdot 1_M$ is then a consequence of the following lemma.