Proposition 9

Let $(E_i)_{i\in I}$ be a family of a-modules, $E=\prod_{i\in I}E_i$ their product. For the a-module $E$ to be injective, it is necessary and sufficient that each of the $E_i$ be so.

Let $u : M' \to M$ be an injective homomorphism of a-modules. For the product homomorphism

to be surjective, it is necessary and sufficient that each of the homomorphisms $\operatorname{Hom}_A(M,E_i)\to\operatorname{Hom}_A(M',E_i)$ be so (II, p. 10, prop. 5); this proves the proposition since $\prod_{i\in I}\operatorname{Hom}_A(M,E_i)$ is canonically identified with $\operatorname{Hom}_A(M,E)$.