Let \(R\) be a ring. An \(R\)-module \(P\) is projective if for every surjective \(R\)-linear map \(g\colon M\to N\) and every \(R\)-linear map \(f\colon P\to N\) there exists an \(R\)-linear map \(\phi\colon P\to M\) with \(g\circ\phi=f\):
An \(R\)-module \(P\) is projective if and only if there are an \(R\)-module \(Q\) and a free \(R\)-module \(F\) with \(F\cong P\oplus Q\).
Suppose \(P\) is projective. Let \(F\) be the free \(R\)-module on the set \(P\) and \(\pi\colon F\to P\) the surjection sending each basis element to the corresponding element of \(P\). Applying the definition to \(g=\pi\) and \(f=\id_P\) gives \(s\colon P\to F\) with \(\pi\circ s=\id_P\), and then \(F\cong P\oplus\ker\pi\) via \(x\mapsto(\pi(x),x-s\pi(x))\).
Conversely, a free module \(F\) with basis \((e_i)_{i\in I}\) is projective: given \(g\colon M\to N\) surjective and \(f\colon F\to N\), choose \(m_i\in M\) with \(g(m_i)=f(e_i)\) and let \(\phi\) be the \(R\)-linear map with \(\phi(e_i)=m_i\). If \(F=P\oplus Q\) is free and \(f\colon P\to N\) is given, lift \(f\oplus 0\colon F\to N\) to \(\psi\colon F\to M\); the restriction \(\phi=\psi|_P\) satisfies \(g\circ\phi=f\).