Corollary 2

Let K be a commutative field, A a K-algebra, and M an A-module. Suppose that for every extension L of K, the $A_{(L)}$-module $M_{(L)}$ is simple. Then the endomorphism ring of M consists of the homotheties $\alpha_M$ with $\alpha \in K$.

Let I be a set of cardinal strictly greater than the dimension of M over K (for example, the set of subsets of M). Let L be an algebraic closure of the field $K((X_i)_{i\in I})$ (V, §4, No. 3, p. 23, Theorem 2). Choose a K-linear form $\varphi$ on L such that $\varphi (1) = 1$, and let $v: M_{(L)}\rightarrow M$ be the K-linear mapping characterized by $v(\alpha \otimes m) =\varphi (\alpha )m$. Let $u$ be an endomorphism of M. The dimension of $M_{(L)}$ over L is equal to that of M over K and is strictly less than the cardinal of L. By Theorem 1, the endomorphism $1_L\otimes u$ of the $A_{(L)}$-module $M_{(L)}$ is of the form $\lambda \otimes 1_M$ with $\lambda \in L$. For every $x\in M$, we have

so that $u$ is the homothety $\varphi (\lambda )_M$.