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Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is local then a projective module is free.

We also have a very big question on free property of projective module over a polynomial ring, that was Serre's conjecture, and now is Quillen-Suslin's theorem.

I wonder, do we have a general condition for a ring $R$ so that every projective $R$-module is free which involves all of the cases mentioned above ?

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    Over a commutative (with unity) semilocal (i.e. finitely many maximal ideals) ring with no non-trivial idempotents , every projective module is free. This is proved in https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://projecteuclid.org/download/pdf_1/euclid.tmj/1178244175&ved=2ahUKEwje_5bv1f_aAhULp48KHYSNA6wQFjAAegQIARAB&usg=AOvVaw0Wg3pDYtBZHdtx_2aqGyrK . This generalizes Kaplansky's theorem for projective modules over local rings.2018-05-12

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