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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 $N \cong \oplus_{j \in J} S_{j}$?

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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.

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