Surjection by Free Module

Theorem
Let $A$ be a ring.

Let $M$ be a left $A$-module.

Then there exists a free $A$-module $F$ and a surjective $A$-module homomorphism $f : F \to M$.

Proof
Let $F = A^{\paren M}$ be the free $A$-module on the set $M$.

Let $f : F \to M$ be the $A$-module homomorphism induced the by the Universal Property of Free Modules applied to the identity $\operatorname {id}_M$ of $M$.

We have
 * $f \circ c = \operatorname {id}_M$

Thus $f$ is a retraction in the category of sets.

By Retraction of Sets is Surjective $f$ is surjective.