As I' m currently dealing with artinian and noetherian modules, I' m asking myself whether there is an artinian module which is not noetherian.
I think so, but even after I thought a lot about this, I didn't find an example. Does anybody find one?
2026-03-26 12:33:35.1774528415
$R$-module $M$ which is artinian, but not noetherian
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Let $p$ be a prime. The Prufer $p$-group $\mathbb{Z}[\frac{1}{p}]/\mathbb{Z}$ is an Artinian $\mathbb{Z}$-module, but is not a Noetherian $\mathbb{Z}$-module.