On Fox quotients of arbitrary group algebras

For a group $G$, N-series $\mathcal{G}$ of $G$ and commutative ring $R$ let $I_{R,\mathcal{G}}^n(G)$, $n\ge 0$, denote the filtration of the group algebra $R(G)$ induced by $\mathcal{G}$, and $I_R(G)$ its augmentation ideal. For subgroups $H$ of $G$, left ideals $J$ of $R(H)$ and right $H$-submodules $M$ of $I_Z(G)$ the quotients $I_R(G)J\mathrm{/}MJ$ are studied by homological methods, notably for $M= I_Z(G)I_Z(H)$,$I_Z(H)I_Z(G) + I_Z([H,G])Z(G)$and$Z(G)I_Z(N) +I^n_{Z,\cal G}(G)$with$N \lhd G$wherethegroup$I_R(G)J/MJ$ is completely determined for $n=2$. The groups $I_{Z,\mathcal{G}}^{n-1}(G)I_Z(H)\mathrm{/}{I}_{Z,\mathcal{G}}^n(G)I_Z(H)$ are studied and explicitly computed for $n\le 3$ in terms of enveloping rings of certain graded Lie rings and of torsion products of abelian groups.