Remark

Let $E$ be an injective a-module, $u : M' \to M$ and $f : M' \to E$ homomorphisms of a-modules. If $\operatorname{Ker}u\subset\operatorname{Ker}f$, there exists a homomorphism $g : M \to E$ such that $g\circ u=f$. This follows indeed from what precedes applied to the injective homomorphism $M'/\operatorname{Ker}u \to M$ deduced from $u$.