The assertion is false. The <module over a ring> $\mathbb Z$ over itself is <Noetherian module>[Noetherian], because its submodules are the <principal ideal>[principal ideals] $n\mathbb Z$, but the descending chain
$$
2\mathbb Z\supsetneq2^2\mathbb Z\supsetneq2^3\mathbb Z\supsetneq\cdots
$$
shows that it is not an <Artinian module>.

Solution

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

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An $R$-module is Artinian when every descending chain of submodules stabilizes. Equivalently, every nonempty family of submodules has a minimal member.


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 Incoming links: Artinian module