Proposition 12

Let M be a module over the semisimple ring A and $\mathscr{S}_M\subset \mathscr{S}$ the support of M. Then the annihilator Ann(M) of M is the two-sided ideal $\sum_{\lambda\in\mathscr{S}-\mathscr{S}_M}\mathfrak{a}_{\lambda}$, and the trace ideal $\tau (M)$ of M is the two-sided ideal $\sum_{\lambda\in\mathscr{S}_M}\mathfrak{a}_{\lambda}$. In particular, A is the direct sum of Ann(M) and $\tau (M)$.

By definition (VIII, p. 84)$,\mathscr{S}_M$ consists of the classes of simple submodules of M. Since the module M is semisimple, the annihilator of M is the intersection of the annihilators $\mathfrak{b}_{\lambda}$ of the modules of class $\lambda$, for $\lambda$ running through $\mathscr{S}_M$. Now, we have $\mathfrak{b}_{\lambda}=\sum_{\mu\not=\lambda}\mathfrak{a}_\mu$ for every $\lambda \in \mathscr{S}$ (Proposition 9). Since A is the direct sum of the family $(\mathfrak{a}_{\lambda})_{\lambda\in\mathscr{S}}$, the annihilator of M is indeed equal to $\sum_{\lambda\in\mathscr{S}-\mathscr{S}_M}\mathfrak{a}_{\lambda}$.

By definition (VIII, p. 80), the trace ideal $\tau (M)$ is the A-submodule of $A_s$ generated by the images of the A-linear mappings from M to $A_s$. Because M is semisimple, it amounts to the same to say that $\tau (M)$ is generated by the simple submodules of $A_s$ with class belonging to $\mathscr{S}_M$. We therefore have $\tau (M) =\sum_{\lambda\in\mathscr{S}_M}\mathfrak{a}_{\lambda}$.