Maximal-submodule splitting criterion for semisimplicity ($\forall M\text{ maximal in }V,V=M\oplus C⟹V\text{ semisimple}$)

For a finite-dimensional module, it suffices that every maximal proper submodule have a submodule complement in order that every submodule have one. Let $W$ be the sum of all simple modules contained in the given module. If $W$ is proper, choose a maximal submodule $M$ containing it. A complement $C$ is isomorphic to the simple quotient $V/M$ and hence is itself simple. This forces $C\subseteq W\subseteq M$, contradicting the complement. Thus $W=V$. Greedily adding simple summands not already in the sum gives a direct sum in finitely many steps. Finally, given any $U$, choose a maximal sum of these simple summands disjoint from $U$. A missing simple summand has zero intersection with their sum plus $U$, contradicting maximality unless their sum complements $U$. This proves the semisimple module criterion, including the zero module.