I have posted this question on MS for 2 weeks however there is no answer up to now. So I post it here and I hope MOers can help me.

Let $\mathfrak{R}$ be a ring, then we knew that a free module over $\mathfrak{R}$ is projective. Moreover, if $\mathfrak{R}$ is a principal ideal domain then a module over $\mathfrak{R}$ is free if and only if it is projective or if $\mathfrak{R}$ is local then a projective module is free.

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

I wonder that, do we have a general condition for a ring $\mathfrak{R}$, so that every projective module over $\mathfrak{R}$ is also free over $\mathfrak{R}$ which involve all of the cases that mentioned above ?