Let $R$ be a commutative, reduced ring. It can be seen that $K_0(R) \cong \mathbb Z$ as groups if and only if every finitely generated projective module is stably free. My question is, are there similar interpretations when one in the following list happens or at least some broad class of rings for which one of the following happens:
(1) $K_0(R)$ is a finitely generated, free abelian group.
(2) $K_0(R)$ is finitely generated.
(3) $K_0(R)$ is free of infinite/uncountable rank.