Exercise 25, § 2

(a) For an $A$-module $Q$ to be injective, it is sufficient that, for every projective $A$-module $P$ and every submodule $P'$ of $P$, every linear mapping of $P'$ into $Q$ can be extended to a linear mapping of $P$ into $Q$ (use the fact that every $A$-module is a quotient of a projective $A$-module).

(b) For an $A$-module $P$ to be projective, it is sufficient that, for every injective $A$-module $Q$ and every quotient module $Q''$ of $Q$, every homomorphism of $P$ into $Q''$ be of the form $\phi \circ u$, where $\phi : Q \to Q''$ is the canonical homomorphism and $u$ is a homomorphism of $P$ into $Q$ (use the fact that every $A$-module is a submodule of an injective $A$-module).