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Is this relation true?

  • $M$ is projective module if and only if $M_m$ is projective module for every maximal ideal $m$.
  • $M$ is finitely generated free module if and only if $M_m$ is finitely generated free module for every maximal $w$-ideal $m$.

Thank you so much

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  • $\begingroup$ What is $M$? What is $M_m$? What is $w$? $\endgroup$ – Kevin Long Nov 15 '17 at 19:05
  • $\begingroup$ I'am so sorry. $M$ and $M_m$ are module and $w$ is operation as star operation. $\endgroup$ – mohammed Nov 15 '17 at 23:56
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Over a commutative Noetherian ring a finitely generated module is projective if and only if it's localization at every maximal ideal is projective (in fact free). Over a non-Noetherian commutative ring it can happen that a finitely generated module is not projective yet every localization is free. So in general the answer to both of your questions is no.

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