On his book, "Algebra: Rings, Modules and Categories", C. Faith gives the next assertion:
$$\mbox{If} \oplus\mbox{ is a }\mbox{-}module\mbox{ thus }\mbox{ and }\mbox{ are injective as }\mbox{-modules}.$$
I've tried to prove this, but i don't know how conclude the injectivity. Since $Q$ is quasi-injective thus $Q$ is both $A$ and $B$-injective, also $A$ and $B$ are relative injective, in particular are $QI$, but since here i don't know how continue...
Any hint?