Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 101 / 1 / a / iv

The assertion is false. For a prime number $p$, the Prüfer p-group $C_{p^{\infty }}$ is an Artinian $\mathbb{Z}$-module: every proper subgroup is a finite cyclic group, so no infinite strictly descending chain of subgroups exists. It is not Noetherian because its cyclic subgroups form the strict ascending chain