Let $f: A \to B$ be an injective map. Then we have a short exact sequence $0 \to A \to B \to coker(f) \to 0 $ where $coker(f)= B/Im(f)$ and $A,B$ are $\mathbb Q$-vector spaces. Is $coker(f)$ a free module?
I know that $A,B$ are free as they are $\mathbb Q$-vector spaces. But I can't argue that $coker(f)$ is free... I need some help...