Manfred Hartl. 
The second Fox quotient of arbitrary group rings. 

Abstract : The second generalized Fox quotient $I(G)J/I^2(G)J$ of the 
augmentation ideal $I(G)$ is determined for
arbitrary groups $G$, subgroups $H$ of $G$ and left ideals $J$ of the 
group ring $Z(H)$, in terms of $G$ and $J$. Moreover,
various related results in the literature are given a unifying easy new 
proof. These results are based on a homological method of
`separating' terms containing $G$ and $H$, which may also serve as a 
basis of studying higher quotients of a similar type.