Using the result that short exact sequence is split exact to prove that module is injective

Prove that The following conditions on a ring R [with identity] are equivalent :

(a) Every [unitary] R-module is projective.

(b) Every short exact sequence of [unitary] R-modules is split exact.

(c) Every [unitary] R-module is injective.

This is exercise 1 in Hungerford's Algebra (page 198). The solution is a direct consequence of Theorem 3.4 (page 192) and Proposition 3.13 (page 197). Let us see it in details.

Proof: (a)$\Rightarrow$(b). Suppose that every [unitary] R-module is projective. Given any short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ of [unitary] R-modules, since $C$ is projective, we have, by Theorem 3.4, that
$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split exact. So, every short exact sequence of [unitary] R-modules is split exact.

(b)$\Rightarrow$(a). Suppose that every short exact sequence of [unitary] R-modules is split exact. Given any R-module $C$, since every short exact sequence of [unitary] R-modules is split exact, we have that every short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split exact. So, by Theorem 3.4, that $C$ is projective. So, every [unitary] R-module is projective.

The equivalence of (b) and (c) follows in a similar way, using Proposition 3.13, instead of Theorem 3.4.

(c)$\Rightarrow$(b). Suppose that every [unitary] R-module is injective. Given any short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ of [unitary] R-modules, since $A$ is injective, we have, by Proposition 3.13, that
$0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split exact. So, every short exact sequence of [unitary] R-modules is split exact.

(b)$\Rightarrow$(c). Suppose that every short exact sequence of [unitary] R-modules is split exact. Given any R-module $A$, since every short exact sequence of [unitary] R-modules is split exact, we have that every short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is split exact. So, by Proposition 3.13, $A$ is injective. So, every [unitary] R-module is injective.