Lemma 3

For the a-module E to be injective, it is necessary and sufficient that, for every injective a-linear mapping $u : M' \to M$, the mapping

be surjective.

If $E$ is injective and if $u : M' \to M$ is injective, then the sequence $0 \to M' \xrightarrow{u} M$ is exact, and hence so is the sequence $\operatorname{Hom}(M,E) \xrightarrow{\operatorname{Hom}(u,1)} \operatorname{Hom}(M',E) \to 0$, and $\operatorname{Hom}(u,1)$ is surjective. Conversely, consider the exact sequence (19); put $M''_1=\nu(M)$ and let $i : M''_1 \to M''$ be the canonical injection and $p : M \to M''_1$ the mapping $m\mapsto \nu(m)$. The sequence $M' \xrightarrow{u} M \xrightarrow{p} M''_1 \to 0$ is exact; by II, p. 36, th. 1, the sequence

is exact. Moreover, one has $\operatorname{Hom}(\nu,1)=\operatorname{Hom}(p,1)\circ\operatorname{Hom}(i,1)$. If $E$ satisfies the condition of the lemma, $\operatorname{Hom}(i,1)$ is surjective, hence the image of $\operatorname{Hom}(\nu,1)$ is also that of $\operatorname{Hom}(p,1)$, and the sequence (20) is exact.