Proposition 8

a) The mapping $z\mapsto (z_{S_{\lambda}})_{\lambda\in\mathscr{S}_M}$ is an isomorphism from Z to the product of the fields $Z_{\lambda}$.

b) The A-module M is isotypical and nonzero if and only if Z is a field.

c) Let Λ be a subset of $\mathscr{S}_M$. Denote by $e_{\Lambda}$ the unique element of Z such that $(e_{\Lambda})_{S_{\lambda}}= 1_{S_{\lambda}}$ for $\lambda \in \Lambda$ and $(e_{\Lambda})_{S_{\lambda}}= 0$ for $\lambda \in \mathscr{S}_M-\Lambda$. We have $(e_{\Lambda})_{M_{\lambda}}= 1_{M_{\lambda}}$ for $\lambda \in \Lambda$ and $(e_{\Lambda})_{M_{\lambda}}= 0$ for $\lambda \in \mathscr{S}_M-\Lambda$.

d) If the support $\mathscr{S}_M$ of M is finite, then the mapping $\Lambda \mapsto e_{\Lambda}Z$ is a bijection from the set of subsets of $\mathscr{S}_M$ to the set of ideals of Z, and the mapping $\mathfrak{a}\mapsto \mathfrak{a}M$ is a bijection from the set of ideals of Z to the set of $(A,B^o)$-sub-bimodules of M. These bijections are isomorphisms of ordered sets. The reverse bijection sends an $(A,B^o)$-sub-bimodule N of M to the ideal consisting of the elements $z$ of Z that send N into M.

For $\lambda \in \mathscr{S}_M, Z_{\lambda}$ is the common center of the commutant $D_{\lambda}$ and the bicommutant $C_{\lambda}$ of the A-module $S_{\lambda}$. By Proposition 7, c) above, the mapping $c\mapsto (c_{\lambda})_{\lambda\in\mathscr{S}_M}$ is an isomorphism from C to $\prod_{\lambda\in\mathscr{S}_M}C_{\lambda}$. By restriction to the centers, we obtain the isomorphism $z\mapsto (z_{S_{\lambda}})_{\lambda\in\mathscr{S}_M}$ from Z to $\prod_{\lambda\in\mathscr{S}_M}Z_{\lambda}$, whence a).

The ring $\prod_{\lambda\in\mathscr{S}_M}Z_{\lambda}$ is a field if and only if the set $\mathscr{S}_M$ has a single element, whence b).

Assertion c) follows from Proposition 7, a). Suppose that $\mathscr{S}_M$ is finite. It follows from a) and Proposition 8 of I, §8, No. 10, p. 109, that the mapping $\Lambda \mapsto e_{\Lambda}Z$ is an isomorphism of ordered sets from $\mathfrak{P}(\mathscr{S}_M)$ to the set of ideals of Z. Let Λ be a subset of $\mathscr{S}_M$. By c), we have the relation $e_{\Lambda}$ZM $=e_{\Lambda}M =$ $\bigoplus_{\lambda\in\Lambda}M_{\lambda}$; because of Proposition 6, c) of VIII, p. 86, it remains to describe the inverse bijection. But $z\in Z$ sends M into $\bigoplus_{\lambda\in\Lambda}M_{\lambda}$ if and only if $z=e_{\Lambda}z$, that is, $z\in e_{\Lambda}Z$.