每日定理2

Isaacs, $\textit{Character Theory of Finite Groups}$, Corollary(1.6)

Let $F$ be algebraically closed, $A$ an $F$-algebra, and $V$ an irreducible $A$-module. Then $E_A(V)=F\cdot1$, the set of scalar multiplications on $V$.

Pf: Let $\vartheta\in E_A(V)$, $\lambda$ an eigenvalue of  $\vartheta$,

• $E_A(V)$ is a division algebra
• $\vartheta-\lambda\cdot1\in E_A(V)$ and is not invertible

Thus $\vartheta-\lambda\cdot1=0$, and we are done.