Proposition 8

a) For every $\lambda \in \mathscr{S},S_{\lambda}$ is a finite-dimensional right vector space over the field $D_{\lambda}$. When passing to the quotient, the mapping $a\mapsto a_{S_{\lambda}}$ defines a ring isomorphism from $A/\mathfrak{b}_{\lambda}$ to End$_{D_{\lambda}}(S_{\lambda})$.

b) For every $\lambda \in \mathscr{S}$, the ring $A/\mathfrak{b}_{\lambda}$ is simple, and the canonical homomorphism $\psi$ from A to $\prod_{\lambda\in\mathscr{S}}A/\mathfrak{b}_{\lambda}$ is a ring isomorphism.

The ring $A/\mathfrak{b}_{\lambda}$ is isomorphic to $A_{S_{\lambda}}$. By Lemma 1 of VIII, p. 139, this ring is simple. The given mapping from $A/\mathfrak{b}_{\lambda}$ to End$_{D_{\lambda}}(S_{\lambda})$ can be identified with the mapping from $A_{S_{\lambda}}$ to $A''_{S_{\lambda}}$. By Proposition 5 of VIII, p. 139, it is an isomorphism.

The A-module $A_s$ is semisimple, faithful, and balanced. The homomorphism $\psi$ can be identified with the morphism from the bicommutant of $A_s$ to $\prod_{\lambda\in\mathscr{S}}$ End$_{D_{\lambda}}(S_{\lambda})$, which is an isomorphism (VIII, p. 86, Proposition 7, c)).

The simple ring $A/\mathfrak{b}_{\lambda}$ is called the simple factor of A of type $\lambda$.