submodule of a semisimple module has complement

Question

I want to prove that a submodule of a semisimple module admits a complement. That is, if S $\subset$ M, then there exists T $\subset$ M, such that M = S $\bigoplus$ T. One supposes that M = $\bigoplus$ Mi, with Mi simple submodules. Then one may consider those i such that Mi $\cap$ S = {0},say i$\in$ J and take as T := $\bigoplus$Mi, i $\in$J. Since S = $\bigoplus$ (S $\cap$Mi), i $\notin$ J,(since for i $\in$ J, S $\cap$ Mi = {0} then clearly M = S $\bigoplus$ T.

Answer 1

Hint: Either $S$ contains $M_i$ completely or $S \cap M_i=\{0\}$ for all $i$. What happens if $S \neq \sum_{j \in J} M_j$ where $J \subseteq I$ and $M_j \in S$ for all $j \in J$.