Talk:Artinian module

Why does every (nontrivial) descending chain of submodules of $\mathbb {Q} /\mathbb {Z}$ has the form $\langle 1/n_{1}\rangle \supseteq \langle 1/n_{2}\rangle \supseteq ...$, 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)