= Group rings of poly-(cyclic or finite) groups are Noetherian
{title2=$A[G]\text{ is Noetherian}$}
If $A$ is a <left Noetherian ring> and a <right Noetherian ring>, and $G$ is a <poly-(cyclic or finite) group>, then $A[G]$ satisfies both chain conditions. Induct along the <subnormal series>. A finite factor gives a finite free extension on each side, handled by <finite-module extension preserves Noetherianity>. An infinite cyclic factor gives a <skew Laurent polynomial ring>, with the twisting <automorphism> induced by conjugation by a lift of the cyclic generator. The <noncommutative Hilbert basis theorem> and clearing negative powers prove the required chain conditions at that step. In particular $\mathbb Z[G]$ is <Noetherian>.
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