Proposition 11

Let B be a ring, F a B-module and P a (B, A)-bimodule. If F is an injective B-module and P a flat A-module, $\mathrm{Hom}_B(P, F)$ is an injective A-module.

Let $u : M' \to M$ be an injective homomorphism of A-modules. We have a commutative diagram

where $\beta$ and $\beta'$ are the canonical isomorphisms of II, p. 74. Since P is flat over A, the homomorphism $1_P \otimes u : P \otimes_A M' \to P \otimes_A M$ is injective. Since F is injective, $\mathrm{Hom}(1_P \otimes u, 1_F)$ is surjective, hence also $\mathrm{Hom}_A(u, 1)$, which proves that $\mathrm{Hom}_F(P, F)$ is an injective A-module (X, p. 16, lemma 3).