Proposition 10

Denote the center of A by Z. For $\lambda \in \mathscr{S}$, let $Z_{\lambda}$ be the center of the field $D_{\lambda}$.

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

b) Order the set $\mathscr{I}_Z$ of ideals of Z and the set $\mathscr{B}_A$ of two-sided ideals of A by inclusion. The mapping $\mathfrak{a}\mapsto \mathfrak{a}A$ is an isomorphism of ordered sets from $\mathscr{I}_Z$ to $\mathscr{B}_A$. The inverse isomorphism sends a two-sided ideal $\mathfrak{b}$ of A to the ideal $\mathfrak{b}\cap Z$ of Z.

This proposition follows from Proposition 8 of VIII, p. 87 applied to the A-module $A_s$, whose bicommutant is A.