Surjection by Free Module
Jump to navigation
Jump to search
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 $c : M \to A^{\paren M}$ be the canonical mapping on $F$.
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 split epimorphism in the category of sets.
By Split Epimorphism is Epic and Surjection iff Epimorphism in Category of Sets $f$ is surjective.
$\blacksquare$