Proposition 6

Let M be a finitely generated semisimple A-module. Denote its support (VIII, p. 66) by $\mathscr{S}_M$ and its endomorphism ring by B. For every $\lambda \in \mathscr{S}_M$, choose a simple A-module $S_{\lambda}$ of class $\lambda$, and denote the left B-module Hom$_A(S_{\lambda},M)$ by $V_{\lambda}$.

a) The ring B is semisimple.

b) The mapping $\lambda \mapsto$ cl(V$_{\lambda}$) is a bijection from the support $\mathscr{S}_M$ of M to the set of classes of simple B-modules.

c) For every $\lambda \in \mathscr{S}_M$, the isotypical component of type $\lambda$ of the A-module M is equal to the isotypical component of type $V_{\lambda}$ of the B-module M.

Viewed as a B-module, M is semisimple (VIII, p. 85, Proposition 5) and faithful. Its countermodule is finitely generated because we have $A_M\subset$ End$_B(M)$. Hence, the ring B is semisimple (Proposition 4). If $(x_1, . . . , x_r)$ is a generating sequence of the A-module M, then the mapping $b\mapsto (bx_1, . . . , bx_r)$ from $B_s$ to $M^r$ is B-linear and injective. Every simple B-module is isomorphic to a submodule of $B_s$ (VIII, p. 136, Proposition 1), hence to a B-submodule of M. The proposition then follows immediately from Proposition 5 of VIII, p. 85.