Ranks of Kn and Gn of orders and group rings of finite groups over integers in number fields

AbstractLet R be the ring of integers in a number field F, Λ any R-order in a semi-simple F-algebra σ, Γ any maximal R-order containing Λ. We show in this paper that for all n ≥ 2 rank Kn(Λ) = rank Gn(Λ) = rank Kn(Γ) = rank Kn(σ). Hence if G is a finite group, rank Kn(RG) = rank Gn(RG) = rank Kn(FG)