Let \(R\) be a ring. An $R$-module \(P\) is projective if for every surjective \(R\)-linear map \(g\colon N \to M\) and every \(R\)-linear map \(f\colon P\to M\) there exists an \(R\)-linear map \(\tilde f\colon P\to N\) with \(g\circ\tilde f = f\):
Let \(R\) be a ring and \(P\) an \(R\)-module. The following are equivalent:
\(P\) is projective;
\(P\) is a direct summand of a free module: there are a free \(R\)-module \(F\) and a submodule \(T\leq F\) with \(F = P \oplus T\), up to isomorphism;
every short exact sequence \(0\to A\to B\xrightarrow{p} P\to 0\) of \(R\)-modules splits: there is an \(R\)-linear map \(s\colon P\to B\) with \(p\circ s = \id_P\);
the functor \(\Hom_R(P, \wait)\) from \(R\)-modules to abelian groups is exact.
(1)\(\Rightarrow\)(3): lift \(\id_P\) along the surjection \(p\). (3)\(\Rightarrow\)(2): choose a free module \(F\) with a surjection \(p\colon F\to P\), for example on a generating set of \(P\); a section \(s\) of \(p\) gives \(F=s(P)\oplus\ker p\) with \(s(P)\cong P\). (2)\(\Rightarrow\)(1): a free module \(F\) with basis \((e_i)\) is projective, since \(\tilde f(e_i)\) may be chosen as any preimage under \(g\) of \(f(e_i)\); if \(F=P\oplus T\) with inclusion \(\iota\colon P\to F\) and projection \(\pi\colon F\to P\), a lift \(h\) of \(f\circ\pi\) gives the lift \(\tilde f=h\circ\iota\) of \(f\). (1)\(\Leftrightarrow\)(4): \(\Hom_R(P,\wait)\) is always left exact, and it preserves surjections exactly when every \(f\colon P\to M\) lifts along every surjection \(g\colon N\to M\).
In (3), the section \(s\) gives \(B = s(P) \oplus \ker p\), and \(s(P)\cong P\).