Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(1.14)

Let $A$ be an $F$-algebra. Then every irreducible $A$-module is isomorphic to a factor module of $A^\circ$. If $A$ is semisimple, then every irreducible $A$-module is isomorphic to a submodule of $A^\circ$.

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Pf:

  • Choose $0\neq v\in V$, an irreducible $A$-module
  • Define $\vartheta:A\rightarrow V$ by $\vartheta(x)=vx$ and $\vartheta\in Hom_A(A^\circ,V)$
  • $A^\circ/{ker\vartheta}\cong V$

Isaacs, $\textit{Character Theory of Finite Groups}$, Theorem(1.15)

Let $A$ be a semisimple algebra and let $M$ be an irreducible $A$-module. Then

  1. $M(A)$ is a minimal ideal of $A$
  2. if $W$ is irreducible, then it is annihilated by $M(A)$ unless $W\cong M$
  3. the map $x\mapsto x_M$ is one-to-one from $M(A)$ onto $A_M\subseteq End(M)$
  4. $\mathcal{M}(A)$ is a finite set
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