Exercise 25, § 1

Keep the notation of Exercise 24.

a) For any subgroup H of G, let $I_H$ be the kernel of the canonical surjection $\mathbf{Z}[G]\rightarrow \mathbf{Z}^{(G/H)}$. Then the mapping $H\mapsto I_H$ is the increasing injection from the set of subgroups of G to the set of the left ideals of $\mathbf{Z}[G]$; if H is a normal subgroup of G, the ideal $I_H$ is two-sided.

b) Suppose that the ring A[G] is left Noetherian. It is then also right Noetherian. For every subgroup H of G, the ring A[H] is left Noetherian; in particular, the ring A is Noetherian. If H is normal in G, then the ring $A[G/H]$ is left Noetherian. Every subgroup of G admits a finite generating system.

c) Prove that the following properties are equivalent:

(i) The group G admits a composition series whose quotients are finite or isomorphic to $\mathbf{Z}$.

(ii) The group G admits a composition series $(G_i)_{0\leqslant i\leqslant n}$ such that $G_1$ has finite index in G and $G_i/G_{i+1}$ is isomorphic to $\mathbf{Z}$ for $0\leqslant i\leqslant n-1$.

When they are satisfied and the ring A is Noetherian, the ring A[G] is left Noetherian (use Exercise 22).