Hopkins-Levitzki theorem: a left Artinian ring is left Noetherian.
I find this quite amazing. I find this asymmetry shocking. It just seems plain unreasonable that there are rings where every ascending chain of ideals stabilizes but not every descending chain stabilizes, and at the same time every ring with stabilizing descending chains has stabilizing ascending chains.
I know asymmetries abound in ring/module theory, but this one strikes me as more elementary and uncanny.
My question is:
Why does this happen?
Of course, this question is at an informal level; I'm not asking for a proof of the theorem. I just want to understand why one chain condition implies the other, but not the other way around. At first glance, it just seems so symmetrical, that I would have expected the conditions to be equivalent, or to have neither condition implying the other.
My very naive first observation is that for noetherian rings, we have the characterization "every ideal is finitely generated", but for artinian rings there is not (that I know of) a simple analog, which is perhaps the first spark of an asymmetry...