$R$-module $M$ which is artinian, but not noetherian

Question

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?

https://en.wikipedia.org/wiki/Artinian_module#Relation_to_the_Noetherian_condition — Qiaochu Yuan, Jul 11 '16 at 21:41

carmichael561 · Answer 1 · 2016-07-11T21:41:51.417

1

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.

edited Jul 11 '16 at 21:41

answered Jul 11 '16 at 21:35

carmichael561

53,688

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This is also example 6.3 of Atiyah-MacDonald (for reference), although formulated as a subgroup of $\Bbb Q/\Bbb Z$. – Arthur Jul 11 '16 at 21:45
Thank you very much! This is really nice. – Peter123 Jul 11 '16 at 21:51

$R$-module $M$ which is artinian, but not noetherian

1 Answers1