Exercise 22, § 1

Let G be a group and H a normal subgroup of G such that the quotient group $G/H$ is isomorphic to $\mathbf{Z}$. Let $g$ be an element of G whose canonical image in $G/H$ generates $G/H$. Let C be a commutative ring, A the algebra C[H] of H over C (III, §2, No. 6, p. $446$)$, (e_h)_{h\in H}$ its canonical basis, and $\sigma$ the automorphism of A that sends $e_h$ to $e_{ghg^{-1}}$.

a) Prove that there exists a unique homomorphism $u$ from the ring $B = A[X]_{\sigma}$ to the ring C[G] that extends the canonical injection of A = C[H] into C[G] and sends X to the element $e_g$ of the canonical basis of C[G].

b) Set $S =\{X\}$. Prove that $u$ extends to an isomorphism from the ring $S^{-1}B$ (cf. Exercise 21) to C[G]. Deduce that if the ring C[H] is left Noetherian, then so is the ring C[G].