Corollary 4

For the $A$-module $E$ to be injective, it is necessary and sufficient that every injective $A$-homomorphism $f:E\to F$ possess an $A$-linear retraction.

Suppose $E$ is injective and let $f:E\to F$ be an injective $A$-homomorphism. Then

is surjective; there therefore exists $r\in\operatorname{Hom}_A(F,E)$ such that $r\circ f=1_E$ and $r$ is an $A$-linear retraction of $f$. Conversely, there exists an injective $A$-module $I$ and an injective $A$-homomorphism $f:E\to I$ (cor. 2); if $f$ possesses an $A$-linear retraction, $E$ is injective by prop. 9 of X, p. 16.