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If $\kappa$ is any cardinal, then one may define a "$\kappa$-Noetherian" ring as a ring such that for any module that has a generating set $S$ satisfying $|S|< \kappa$, then any submodule also has such a generating set. Then a Noetherian ring is just a $\aleph_0$-Noetherian ring with this definition (and a principal ring is a $2$-Noetherian domain, but I think that probably only infinite cardinals are really meaningful in this setup). I you prefer non-commutative rings, you may add "left" and "right" wherever needed.

Question 1a: Does such a notion exist somewhere ? Is it interesting, or even useful ?

My main question is based on the following observation : if on the other hand you want to generalize the ascending chain condition, then you will naturally get a condition on ordinals, since it is formulated in terms of an order. Namely, you can ask wether there are chains of ideals having the order structure of a given ordinal : if $\alpha$ is an ordinal, a ring $R$ will be said to satisfy $\alpha$-$(AC)$ if there is no strictly increasing function $\alpha \to I(R)$ where $I(R)$ is the set of ideals, given the inclusion order. Then a Noetherian ring is just a ring satisfying $\omega$-$(AC)$. Again, you can add "left" or "right" if you want to.

Question 1b: Same as question 1, for this other notion.

So then there are two quite objectively natural notions that generalize Noetherian rings, but one is based on cardinals, and the other on ordinals. There is a sort of "coincidence" in the fact that $\omega$ and $\aleph_0$ are pretty much equivalent in the sense that finite ordinals and finite cardinals are really the same thing (same objects, same operations, etc.). But this will no longer be true for higher cardinals and ordinals.

Question 2: Should one be considered the right one ? If so, which one and why ?

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  • $\begingroup$ I think I've seen $\aleph_0$-Noetherian used to mean every ideal is countably generated, which seems related to, and maybe equivalent to, your $\aleph_1$-Noetherian (which is a better name!). $\endgroup$ – Jeremy Rickard Mar 8 '16 at 15:08
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The thing about chains is that they have cofinal "cardinal chains". Namely, if $\alpha$ is any countable limit ordinal, then there is an $\omega$ sequence which is unbounded in $\alpha$.

If $\alpha$ is a limit ordinal of cardinality $\aleph_1$, then either there is an unbounded chain of order type $\omega$ or there is an unbounded chain of order type $\omega_1$ (but never both when the axiom of choice is assumed).

And so on. If $\alpha$ is a limit ordinal of cardinality $\kappa$, then there is a cardinal $\lambda\leq\kappa$, and an unbounded chain of order type $\lambda$.

So even if you talk about countable order type, it suffices to talk about $\omega$. And so on. So there is really no confusion between the ordinal and cardinal versions.

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