Talk:Artinian module

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

I propose therefore to replace $\mathbb{\mathbb{Q}}/\mathbb{\mathbb{Z}}$ by $\mathbb{\mathbb{Z}}(p^{\mathrm{\infty }})$ 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)