Let R be a commutative ring with 1, G an Abelian group, RG the group ring on R and G. In this paper we gave some properties of PF- rings in which f. g. projective modules are free. The Grothendieck groups K0(RG) for some cases are given. In addition, for the ring R with the unimodular column property, we proved the following result: K0(RG) ≈K0(R), hence if R ∈PF, then K0(RG)≈Z . |