Corollary

Suppose that A is an algebra over an algebraically closed commutative field K and that M is a semisimple A-module that is finite-dimensional as a vector space over K. For every $\lambda$ in $\mathscr{S}_M$, denote by $e_{\lambda}$ the projector of M with image $M_{\lambda}$ and kernel $\oplus_{\lambda\not=\mu}M_\mu$. Then $(e_{\lambda})_{\lambda\in\mathscr{S}_M}$ is a basis of the vector space Z over K.

Since M is a finite-dimensional vector space over K that is the direct sum of the family of nonzero submodules $(M_{\lambda})_{\lambda\in\mathscr{S}_M}$, the set $\mathscr{S}_M$ is finite, and each of the spaces $S_{\lambda}$, for $\lambda \in \mathscr{S}_M$, is finite-dimensional over K. Since the field K is algebraically closed, we have $D_{\lambda}= Z_{\lambda}= K$ (VIII, p. 47, Theorem 1), and the mapping $z\mapsto (z_{S_{\lambda}})_{\lambda\in\mathscr{S}_M}$ is an isomorphism from Z to $K^{\mathscr{S}_M}$ (Proposition 8, a)). The corollary then follows from part c) of Proposition 8.