Theorem 1

Let M be a module that is the (not necessarily direct) sum of a family $(S_i)_{i\in I}$ of simple submodules, and let N be a submodule of M. There exists a subset J of I such that M is the direct sum of the family consisting of N and the modules $S_j$ for $j$ running through J.

Let $\mathscr{S}$ be the set of subsets $I'$ of I such that the sum of the family consisting of the modules N and $S_i$ for $i$ in $I'$ is direct. The set $\mathscr{S}$ is of finite character: a subset J of I belongs to $\mathscr{S}$ if and only if the same holds for every finite subset of J. Hence, the set $\mathscr{S}$ has a maximal element J (Set Theory, III, §4, No. 5, p. 171). Set $N'= N +\sum_{j\in J}S_j$. Let $i$ be in I - J. Since J is maximal in $\mathscr{S}$, the set $J\cup \{i\}$ does not belong to $\mathscr{S}$, so that $S_i\cap N'\not= 0$. Since $S_i$ is a simple module, we have $S_i\cap N'= S_i$. We therefore have $S_i\subset N'$ for every $i\in I$, whence $N'= M$. This completes the proof.