Corollary

Let A be a ring and $\mathscr{S}$ the set of classes of simple A-modules. Let M be a semisimple A-module and N a submodule of M. Then we have $N =\bigoplus_{\lambda\in\mathscr{S}}N\cap M_{\lambda}$ and $M/N =\bigoplus_{\lambda\in\mathscr{S}}(M_{\lambda}+ N)/N$.

Since N and $M/N$ are semisimple (VIII, p. 56, Corollary 3), the corollary follows from Proposition 4, d).

The support of a semisimple A-module M is the set of classes $\lambda$ of simple A-modules such that the isotypical component of M of type $\lambda$ is nonzero. The support of a finitely generated semisimple A-module is finite.