Let $R$ be a Noetherian ring which has a identity element and $M$ be a finitely generated $R$ module.
We write $M_\mathfrak{p}$ for the localization of $M$ with respect to prime ideal $\mathfrak{p}$ of $R$. So does $R_\mathfrak{p}$.
And we define $R_f = R[\frac{1}{f}],\ M_f = M \otimes_R R_f$.
On Assuming (1), we want to prove (2). ((1) $\Rightarrow$ (2))
(1) $\forall \mathfrak{p} \in Spec(R),\ \exists f \in R- \mathfrak{p}$ s.t. $M_f$ is $R_f$ free module.
(2) $\forall \mathfrak{p} \in Spec(R),\ $ $M_\mathfrak{p}\ \ $is$\ R_\mathfrak{p}\ $free module.
But I can't do that. Please give me some help.