Talk:Artinian module

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Geometry guy 18:52, 21 May 2007 (UTC)

Why does every (nontrivial) descending chain of submodules of $\mathbb{Q} / \mathbb{Z}$ has the form $\langle1/ n_1 \rangle \supseteq \langle 1/ n_2 \rangle \supseteq ...$ , like in the article? This is clear for the quasicyclic groups $\mathbb{Z}(p^{\infty})$ , for each prime $p$ , because every proper subgroup is finite and cyclic. But this does not hold for $\mathbb{Q} / \mathbb{Z}$ . Where's the information from, that $\mathbb{Q} / \mathbb{Z}$ is artinian as $\mathbb{Z}$ -module? I have a little doubt about that ;-)

I propose therefore to replace $\mathbb{Q} / \mathbb{Z}$ by $\mathbb{Z}(p^{\infty})$ in the article. This gives certainly a counterexample for artinian => noetherian for modules. ---oo- 07:12, 16 September 2005 (UTC)

Since nobody vetoes, I've just corrected it. ;-) ---oo- 16:11, 18 September 2005 (UTC)

References?--Cronholm¹⁴⁴ 04:21, 18 July 2007 (UTC)

Added. I don't know where the counterexample comes from, though. Ryan Reich 13:19, 18 July 2007 (UTC)

Talk:Artinian module

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