Finitely generated flat modules and a characterization of semiperfect rings

For a ring S, let K0(FGFl(S)) and K0(FGPr(S)) denote the Grothendieck groups of the category of all finitely generated flat 5-modules and the category of all finitely generated projective 5-modules respectively. We prove that a semilocal ring R is semiperfect if and only if the group homomorphism K0(FGFl(R)) → K0(FGFl(R/J(R))) is an epimorphism and K0(FGFl(R)) = K0(FGPr(R)).