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One of the first results I learned about chain conditions is that Artinian rings (satisfying d.c.c.) are automatically Noetherian (satisfying a.c.c.), and in fact a ring is Artinian if and only if it's Noetherian and has Krull dimension $0$, a very strong condition.

My question is: why are these so asymmetric? Just by looking at the chain conditions themselves you would think that they're "dual" or something, but it seems like being Artinian is such a strong condition they're not even bothered to be studied that much (by which I mean, skimming the table of contents of a few algebra books I can't even see it show up).

I'm dimly aware that this doesn't hold for modules (though this is outside the scope of what I'm currently studying) -- perhaps one way to answer my question is to explain why rings and modules differ in this respect.

fish
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  • That doesn't explain everything, but one possible (part of an) answer could be the following : $\mathbb Z$, the initial ring, is not artinian. $\mathbb Z[X_1,...,X_n]$ is not artinian either. So we have this asymmetry at the level of the most basic rings (the free rings on finitely many generators). If you can understand that one, then you'll be onto something (for those rings, the reason is "it's easier to divide someone, than be divided by them") – Maxime Ramzi Mar 31 '20 at 15:02

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