Proposition 7

Let M be an A-module. The canonical mapping $\alpha_M$ is injective. For every $\lambda \in \mathscr{S}$, the image under $\alpha_M$ of $S_{\lambda}\otimes_{D_{\lambda}}$ Hom$_A(S_{\lambda},M)$ is the isotypical component of M of type $\lambda$. The image of $\alpha_M$ is the socle of M. The A-module M is semisimple if and only if the mapping $\alpha_M$ is bijective.

Let $\lambda \in \mathscr{S}$. Denote the isotypical component of M of type $\lambda$ by $M_{\lambda}$. Every A-linear mapping from $S_{\lambda}$ to M takes its values in $M_{\lambda}$ (VIII, p. 66, Proposition 5). Consequently, by Proposition 3, a) of VIII, p. 62, the mapping $\alpha_M$ induces a bijection from $S_{\lambda}\otimes_{D_{\lambda}}$ Hom$_A(S_{\lambda},M)$ to $M_{\lambda}$. The proposition follows because the socle of M is the direct sum of the family $(M_{\lambda})_{\lambda\in\mathscr{S}}$ and the module M is semisimple if and only if it is equal to its socle.