Let $R$ be an associative ring with identity, and let $P$ be a unital left $R$-module. The following conditions are equivalent.

1. $P$ is projective; that is, for every surjective $R$-module homomorphism $q: M \to N$ and every $R$-module homomorphism $f: P \to N$, there exists an $R$-module homomorphism $\widetilde{f}: P \to M$ such that $q \circ \widetilde{f} = f$. 2. Every surjective $R$-module homomorphism $q: M \to P$ admits an $R$-linear section, meaning there exists an $R$-module homomorphism $s: P \to M$ such that $q \circ s = \operatorname{id}_P$. 3. There exist a free left $R$-module $F$ and a left $R$-module $Q$ such that $F \cong P \oplus Q$. 4. For every surjective $R$-module homomorphism $q: M \to N$, the induced group homomorphism

\begin{align*} q_*: \operatorname{Hom}_R(P, M) &\to \operatorname{Hom}_R(P, N) \\ \varphi &\mapsto q \circ \varphi \end{align*}

is surjective.