Corollary 2

A module M is semisimple if and only if every submodule of M is a direct factor.

The condition is necessary by Theorem 1.

Conversely, suppose that every submodule of M admits a supplement. Let $M'$ be the sum of the simple submodules of M, and let $M''$ be a supplement of $M'$ in M. Suppose that we have $M'\not= M$ and therefore $M''\not= 0$. Let N be a nonzero monogenous submodule of $M''$. By Proposition 3 of VIII, p. 49, there exists a maximal submodule P of N. Let Q be a submodule supplementary to P in M. Then $N\cap Q$ is a submodule of N, supplementary to P in N, hence isomorphic to $N/P$ (II, §1, No. 9, p. 210, Proposition 13). Consequently, $N\cap Q$ is a simple submodule of $M''$, contrary to the definition of $M'$.

We therefore have $M'= M$, and the module M is semisimple by Corollary 1.