Corollary 2

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

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