Talk:Artinian module

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)