This question is motived by this recent question.

$K_{0}(R)=\mathbb{Z}$ is often used as a euphemism for saying that every finitely generated projective module is stably free; however, there are some subtleties involved.

The statement that every finitely generated projective module is stably free is equivalent to saying that $K_{0}(R)$ is generated by the isomorphism class of $R$. To show that the two are not the same, consider $R = M_{2}(\mathbb{C})$, $2\times 2$ matrices over the complex numbers. Devissage tells us that $K_{0}(R)=\mathbb{Z}$, with the unique simple module as a generator. However, every stably free projective has to have even length as an $R$ module, as length($R$)=2, and thus the unique simple module (projective as $R$ is semisimple) is not stably free.

Are there any commutative examples of this phenomena? More precisely, is there a commutative ring with $K_{0}(R)=\mathbb{Z}$, but with f.g. projectives which aren't stably free? A commutative noetherian one?