The question is:
Let $R$ be an integral domain, and $K(R)(\neq R)$ be the fractional field of $R$. Prove that any non-zero $K(R)$-vector space cannot be a free $R$-module.
Can anybody give a sketch of the proof? What method should I go by to prove this? I'm unable to begin.
Hint: In a free $R$-module, if $v$ is among the generators and $r$ is not a unit, then there is no element $u$ such that $ru=v$.