11.29 (1990)

Let $F$ be a free group and $\mathfrak{f} = \mathbb{Z}F(F - 1)$ the augmentation ideal of the integral group ring $\mathbb{Z}F$. For any normal subgroup $R$ of $F$ define the corresponding ideal $\mathfrak{r} = \mathbb{Z}F(R - 1) = {}_\operatorname{id}(r - 1 \mid r \in R)$. One may identify, for instance, $F \cap (1 + \mathfrak{rf}) = R'$, where $F$ is naturally imbedded into $\mathbb{Z}F$ and $1 + \mathfrak{rf} = \{1 + a \mid a \in \mathfrak{rf}\}$.

Identify in an analogous way in terms of corresponding subgroups of $F$:
$\qquad$ a) $F \cap (1 + \mathfrak{r}_1\mathfrak{r}_2 \cdots \mathfrak{r}_n)$, where $R_i$ are normal subgroups of $F$, $i = 1, 2, \dots, n$;
$\qquad$ b) $F \cap (1 + \mathfrak{r}_1\mathfrak{r}_2\mathfrak{r}_3);$
$\qquad$ c) $F \cap (1 + \mathfrak{fs} + \mathfrak{f}^n)$, where $F/S$ is finitely generated nilpotent;
$\qquad$ d) $F \cap (1 + \mathfrak{fsf} + \mathfrak{f}^n)$;
$\qquad$ e) $F \cap (1 + \mathfrak{r}(k) + \mathfrak{f}^n)$, $n > k \geqslant 2$, where $\mathfrak{r}(k) = \mathfrak{rf}^{k-1} + \mathfrak{frf}^{k-2} + \dots + \mathfrak{f}^{k-1}\mathfrak{r}$;
$\qquad$ f) Is the quotient group $(F \cap (1 + \mathfrak{r} + \mathfrak{f}^n))/R \cdot \gamma_n(F)$ always abelian?