# A Projective Module that is not Free

## A Projective Module that is Not Free

We work over a fixed commutative unital ring $R.$ An $R$-module is projective if given any surjection $f : M \to N$ and map $g : P \to N$ there exists a (not necessarily unique) lift $h : M \to P$ such that $g = fh.$

Proposition. All free modules are projective.

Pf. We need the axiom of choice. Lift the image of each generator $f(e_i)$ to some $m_i$ if the fiber over $f(e_i).$ Q.E.D.

The converse is not true in general.

Counterexample. $R \times \{0\}$ is projective but not free (as a $R$-module).