TITLE: Hopkins-Levitzki: an uncanny asymmetry?
QUESTION [22 upvotes]: Not every left Noetherian ring is left Artinian. Take $\mathbb{Z}$ as a quick example.
But:

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...

REPLY [2 votes]: I transplanted this from a question I later marked as a duplicate of this one.
There is a sharper version of the Hopkins-Levitzki-Akizuki theorem which explains why Noetherian=Artinian over semiprimary rings.
The upshot is that since $R/J(R)$ is semisimple, you can construct a candidate for a composition series for $M$ (at first you do not know if it is finite or not), and then since $J(R)$ is nilpotent, the series is finite, so that you really have a composition series.
Now if a ring is right Artinian, it conveniently implies that $R$ is semiprimary, so that you can conclude right away it is right Noetherian.
But why the asymmetry? Why isn't the other direction true?  I am not able to give a concrete answer, but there is some intuition.
Maybe part of it has to do with identity, or something close to it. After all, you can get a rng that's Artinian but not Notherian if you take such an abelian group and define the product of all pairs to be zero.
Requiring a ring to have identity is a sort of finiteness condition. It's the reason why rings with identity have maximal right ideals, but it does nothing to help find minimal right ideals. In a sense, it is a very weak Noetherian condition: "the collection of proper ideals has a maximal element" as opposed to "every nonempty collection of proper ideals has a maximal element."
It could be that "Artinian + 1" is enough to "catch both up and down" directions of a series with simple factors, so that it has to be finite, hence Noetherian.  Maybe "Noetherian + 1" only work together both to catch the "up" direction of composition series, but they don't do anything for the "down" direction.