Proposition 4

Let A be a ring. Denote the set of classes of simple A-modules by $\mathscr{S}$. Let M be a semisimple A-module.

a) The module M is the direct sum of the family $(M_{\lambda})_{\lambda\in\mathscr{S}}$ of its isotypical components.

b) Suppose that M is the direct sum of a family $(N_i)_{i\in I}$ of simple submodules. For every $\lambda \in \mathscr{S}$, let $I(\lambda )$ be the set of indices $i\in I$ such that $N_i$ is of class $\lambda$. We have $M_{\lambda}=\bigoplus_{i\in I(\lambda)}N_i$.

c) If M is finitely generated, then the set of $\lambda \in \mathscr{S}$ such that $M_{\lambda}\not= 0$ is finite.

d) For every submodule N of M and every $\lambda \in \mathscr{S}$, we have $N_{\lambda}= N\cap M_{\lambda}$ and $(M/N)_{\lambda}= (M_{\lambda}+ N)/N$.

Since M is semisimple, it is the sum of the family $(M_{\lambda})_{\lambda\in\mathscr{S}}$; let us prove that this sum is direct. Let $\lambda \in \mathscr{S}$. Denote the sum of the family $(M_\mu)_{\mu\in\mathscr{S}-\{\lambda\}}$ by $M'_{\lambda}$. The module $M'_{\lambda}$ is the direct sum of a family of simple modules not isomorphic to $\lambda$ (VIII, p. 56, Theorem 1). By Corollary 3 of Theorem 1 of VIII, p. 56, $M'_{\lambda}$ does not contain any simple submodules of class $\lambda$. We consequently have $M_{\lambda}\cap M'_{\lambda}= 0$. Assertion a) is therefore proved. By construction, we have $M_{\lambda}\supset \bigoplus_{i\in I(\lambda)}N_i$, so assertion b) follows from Remark 1 of II, §1, No. 8, p. 208.

Assertion c) follows from a) and Proposition 23 of II, §1, No. 12, p. 221.

Let N be a submodule of M and $\lambda \in \mathscr{S}$. The isotypical component $N_{\lambda}$ of type $\lambda$ of N is contained in $M_{\lambda}$ and $M_{\lambda}\cap N\subset N_{\lambda}$. The intersection $N\cap M_{\lambda}$ is therefore the isotypical component of N of type $\lambda$.

For every $\lambda \in \mathscr{S}$, the module $M_{\lambda}+N/N$ is isomorphic to $M_{\lambda}/(M_{\lambda}\cap N)$. It is therefore isotypical of type $\lambda$ and contained in $(M/N)_{\lambda}$. The last assertion then follows from a) and II, §1, No. 8, p. 208, Remark 1.