Category of Modules has Enough Projectives

Theorem

Let $A$ be a ring.

Then the category of left $A$-modules has enough projectives.

Proof

Let $M$ be an $A$-module.

By Surjection by Free Module there is a free $A$-module $F$ and a surjection $f : F \to M$.

By Epimorphism of Modules Iff Surjection $f$ is an epimorphism.

By Free Module is Projective $F$ is projective.

$\blacksquare$