Some intersections and identifications in integral group rings (Q698358)

This technical paper obtains a number of results on intersections of augmentation ideals in the integral group ring \(\mathbb{Z}[F]\) of a free group \(F\). For example, let \(R\) be a normal subgroup of \(F\), let \(I(R)\) denote the augmentation ideal of \(\mathbb{Z}[R]\), and let \(\gamma_i(R)\) denote the \(i\)-th term of the lower central series of \(R\). Then, for all integers \(n\geq 1\), it is shown here that \(I^{n+1}(F)\cap I^n(R)=I^{n+1}(R)+\sum_{i=1}^nI^{n-i}(R)\cdot I(R_i)\) where \(R_i=\gamma_i(R)\cap\gamma_{i+1}(F)\).