Corollary 1

Let $u : E \to F$ be a linear mapping. For there to exist a linear mapping $v : F \to E$ such that $u \circ v = 1_F$ (the case where $u$ is said to be right invertible and $v$ is said to be the right inverse of $u$), it is necessary and sufficient that $u$ be surjective and that its kernel be a direct factor in $E$. The submodule $\operatorname{Im}(v)$ of $E$ is then a supplement of $\operatorname{Ker}(u)$.

It is obviously necessary that $u$ be surjective; as $v$ is then a section associated with $u$, the conclusion follows from Proposition 15.