The group of normalized units of a group ring (Q1071849)

Let \(RG\) be the group ring of \(G\) over the commutative domain \(R\) of characteristic 0 and let \(\Delta_R(G)\) denote its augmentation ideal. If \(N\) is any normal subgroup of \(G\), let \(\Delta_R(G,N)\) denote the kernel of the natural homomorphism \(RG\to R(G/N)\) so that \(\Delta_R(G,N)=\Delta_R(N)RG\). Finally let \(V(RG)\) be the group of normalized units of \(RG\), that is \(V(RG)=U(RG)\cap (1+\Delta_R(G))\) where \(U(RG)\) is the full group of units. The main result here is as follows. Suppose \(I\) is an ideal of \(RG\) and set \(J=\cap_0^\infty(I+\Delta_R(G))^n\). Then the factor group \[ V(RG)\cap (1+J)/V(RG)\cap (1+\Delta_R(G,G\cap (1+J))) \] is torsion-free. This can sometimes be used to prove that \(G\) has a torsion-free normal complement in \(V(RG)\).