Solution
= Solution
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
$$
C_p\subsetneq C_{p^2}\subsetneq C_{p^3}\subsetneq\cdots.
$$