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As a TA I was recently asked to give the students an introduction to two (quite related) concepts that are new to me, noetherian and artinian modules. I intend to prove the characterisation theorem (i.e., ACC iff every submodule is finitely generated iff maximal submodule condition) and the exact sequences theorem which is demonstrated in Atiyah and MacDonald's book, from which all basic results follow as corollaries.

After showing all this, every source I've consulted either stops talking about them or immediately jumps to noetherian and artinian rings. In both cases, applications aren't discussed. This led me to the natural question,

Why are noetherian and artinian modules important?

I'd love to show the students some applications or examples. Someone even mentioned there is a relation between some remainder theorem (chinese?) and these concepts, is this true?

There's an almost identical question which only deals with the rings case. The most voted answer to that question basically says that noetherian and artinian rings aren't that important and that usually the hypothesis of a ring being either noetherian or artinian can be weakened. I'm not looking for this kind of answers and I would even appreciate non-trivial examples of stronger theorems which also hold for this modules.

Community
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hjhjhj57
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    Well, every finitely generated module over a noetherian ring is automatically noetherian, so maybe that's why you rarely see them discussed. – Zhen Lin Mar 16 '15 at 07:37

2 Answers2

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My main picture of "Artinian" and "Noetherian" is that they are generalized finiteness conditions. (Or even a sort of "smallness" condition, if you like.)

We like things that are "finite" by some measure because they usually are less pathological than the objects that lack the property. You can find many examples in algebraic geometry where the Noetherian condition buys you many nice results.

Just think of how useful the concept of well-ordered sets has been, and then consider that it is just a form of Noetherianness on a totally-ordered set. Noetherian modules and rings are just an application of the same idea in another context.

The Noetherian condition prevents chains from piling up too much, and the Artinian condition prevents them from infinitely shrinking. If you have both conditions on a module at the same time, then you can decompose it into a finite chain of submodules with nice properties (unique in some particular sense.)

vs finite generation

Think about another finiteness condition for a second: finitely generated modules. At first blush, this seems like a natural replacement for "finiteness", just as finite dimensionality is a good replacement for finiteness in vector spaces. But the thing is that finite generation isn't hereditary in modules (that is, submodules of finitely generated modules need not be finitely generated.) When you examine a module all of whose submodules are finitely generated, you are looking at a Noetherian module.

There is a counterpart for the Artinian condition in the form of finitely cogenerated modules which is inherited by quotients instead of submodules.

About rings, even though you are only interested in modules...

It's especially interesting that a right Artinian ring is necessarily right Noetherian. (Of course, this isn't true for modules.) Even assuming this (the stronger of the two conditions) you don't get a particularly strong structure theory about such rings. For commutative Artinian rings you can decompose into a finite product of local Artinian rings, but still local Artinian rings have quite a bit of variety.

Finite dimensional algebras over fields are Artinian, and they have convenient representations in terms of square matrices. If you had a finite dimensional (hence Artinian and Noetherian) module of an $F$-algebra, you could examine its endomorphism ring as a subring of a square matrix ring.

The picture for the structure of right Noetherian rings is even less clear, although there are many special cases.

I guess we should not expect too much though: even the theory of finite rings is pretty wild!

rschwieb
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I think one good example is that they are Serre Subcategories of $R$-Mod

https://ncatlab.org/nlab/show/Serre+subcategory

A Serre subcategory answer the question: Which are conditions necessary and sufficient for a subcategory $\mathcal{S}$ of $R$-Mod for that $\pi\colon R$-Mod$\longrightarrow R$-Mod$/\mathcal{S}$ is an exact functor?

Math.mx
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