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 $V$.

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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.

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