I'm searching for a proof that a finitely generated projective module over a Von Neumann Regular ring such that all the localizations have the same rank is free. I know that this result is true, because a friend of mine have proved it when trying to make an intuitionistic proof of the Serre's conjecture, however he did not publish it and he lost the proof that must be located among the piles of paper in his room.

Thanks in advance.