Proposition 6

a) Viewed as an $(A,B^o)$-bimodule, M is semisimple.

b) For every $\lambda \in \mathscr{S}_M,M_{\lambda}$ is a simple $(A,B^o)$-sub-bimodule of M.

c) For every $(A,B^o)$-sub-bimodule N of M, there exists a unique subset Λ of $\mathscr{S}_M$ such that N is equal to $\oplus_{\lambda\in\Lambda}M_{\lambda}$.

Let $\lambda$ be in $\mathscr{S}_M$. The left A-module $S_{\lambda}$ and the right B-module $V_{\lambda}$ are simple, and $D_{\lambda}$ is the opposite ring of the field of endomorphisms of $S_{\lambda}$. By Corollary 2 of VIII, p. 63, the $(A,B^o$)-bimodule $S_{\lambda}\otimes_{D_{\lambda}}V_{\lambda}$ is simple, and the same holds for $M_{\lambda}$, which is isomorphic to it. This proves b), and a) follows.

If $\lambda$ and $\mu$ are distinct in $\mathscr{S}_M$, then $M_{\lambda}$ and $M_\mu$ are not isomorphic as A-modules, nor a fortiori as $(A,B^o$)-bimodules. Assertion c) follows by Corollary 2 of VIII, p. 68.