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.

  • $\begingroup$ Look in Ken Goodearl's book on von Neumann regular rings. $\endgroup$
    – Nik Weaver
    Commented May 24, 2013 at 3:16
  • 1
    $\begingroup$ Is it true though ? A product of fields is a commutative VNR ring, but you can easily find a principal ideal which is not free.. $\endgroup$
    – Fred.Fred
    Commented May 24, 2013 at 11:54
  • $\begingroup$ Oops, I will delete my wrong answer. Thanks to Fred.Fred and Torsten! $\endgroup$ Commented May 24, 2013 at 21:08
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    $\begingroup$ Ops, I forgot to put that the rank is constant at each localization. $\endgroup$
    – user17868
    Commented Jun 3, 2013 at 14:03
  • $\begingroup$ This holds for all zero-dimensional commutative rings, see my answer here mathoverflow.net/q/323319. $\endgroup$ Commented Feb 15, 2019 at 17:29


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