Show $0 \to A\mathop \to B\mathop \to C \to 0$ is split when $C$ is a free Abelian group

Question

I am looking at a statement that, for a short exact sequence of Abelian groups
$$0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0$$

if $C$ is a free Abelian group, then this sequence is split.

I cannot figured out why, can anybody help?

Answer 1

Hint:

How do you define a homomorphism from a free abelian group (more generally a free $R$-module) into another group/module?

Deduce a homomorphism $s\colon C\to B$ such that $\;g\circ s=\operatorname{id}_C$, and prove that $$B\simeq A\oplus s(C).$$