Definition 5

Let M be a semisimple A-module. A description of M (with respect to the family $(S_{\lambda})_{\lambda\in\mathscr{S}}$) is a pair $((V_{\lambda})_{\lambda\in\mathscr{S}}, \alpha )$, where $V_{\lambda}$ is a left vector space over the field $D_{\lambda}$ for each $\lambda \in \mathscr{S}$ and $\alpha :\bigoplus_{\lambda\in\mathscr{S}}(S_{\lambda}\otimes_{D_{\lambda}}V_{\lambda})\rightarrow M$ is an isomorphism of A-modules.

By Proposition 7, every semisimple A-module M has a canonical description: it is the pair ((Hom$_A(S_{\lambda},M)$)$_{\lambda\in\mathscr{S}}, \alpha_M$), where $\alpha_M$ is the A-linear mapping defined by formula (10).