What is a projective module? Give standard equivalent characterizations.
Solution
An \(R\)-module \(P\) is projective if it has the following lifting property: whenever \[ M\twoheadrightarrow N \] is a surjective \(R\)-linear map and \(f:P\to N\) is any \(R\)-linear map, there exists \(\widetilde f:P\to M\) such that the triangle commutes: \[ P\xrightarrow{\widetilde f}M\twoheadrightarrow N \quad=\quad P\xrightarrow{f}N. \]
The following are equivalent:
- \(P\) is projective.
- Every short exact sequence \[ 0\to A\to B\to P\to0 \] splits.
- \(P\) is a direct summand of a free module: there is a module \(Q\) and a free module \(F\) with \[ F\cong P\oplus Q. \]
- The functor \[ \Hom_R(P,-) \] is exact.
Every free module is projective. Over a PID, every submodule of a free module is free, and every finitely generated projective module is free; over a general ring, projective modules need not be free.