A Projective Module that is Not Free
We work over a fixed commutative unital ring An -module is projective if given any surjection and map there exists a (not necessarily unique) lift such that
Proposition. All free modules are projective.
Pf. We need the axiom of choice. Lift the image of each generator to some if the fiber over Q.E.D.
The converse is not true in general.
Counterexample. is projective but not free (as a -module).