Semisimple submodule

Question

Suppose that $M$ is a semisimple module, i.e $M \cong \oplus_{i \in I} S_{i}$ where $S_{i}$ are simple modules. Let $N$ be a submodule of $M$. Why there exists a subset $J \subseteq I$ such that $N \cong \oplus_{j \in J} S_{j}$?

Answer 1

A submodule of a direct sum of modules is given by specifying a submodule of each direct summand (think about the composition of the inclusion $N\to M$ with the projection maps onto each summand of $M$). In this case, each summand has only two possible submodules.