Category of Modules has Enough Projectives
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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.
$\blacksquare$