Why Auslander and Reiten were so interested in Artin algebras?

Question

I am studying Artin algebras, but I don't understand why they are interested by many scholars? How studying them can help studying Noetherian algebras? Why they are easy to study?

Answer 1

My answer to this question is based on the presumption that you already know why representation theory of finite-dimensional algebras is studied. If not, you should first try to understand the goal and importance of representation theory of finite-dimensional algebras.

Artin algebras are a natural generalization of finite-dimensional algebras and include classes of algebras which are not of this type (e.g. certain factor rings of principal ideal domains and endomorphism rings of finite abelian groups). Thus, artin algebras are more applicable (than merely finite-dimensional algebras) and yet there is little-added complication in their representation theory (compared to finite-dimensional algebras) because artin algebras possess many fundamental "nice'' properties of finite-dimensional algebras; for example,

1. they possess Morita self-duality,
2. the endomorphism rings of their finitely generated modules are again artin algebras,
3. their category of finitely generated modules has enough injective,
4. their category of finitely generated modules are Krull-Schmidt.

As you proceed in your studies, you will realize the importance of these properties, which none of them hold in general over left artinian rings. Even not all of them hold in general over artinian (left and right) rings (property (1) for example). These observations make artin algebras interesting and important for representation theorists.

As for their connection with noetherian rings/algebras mentioned in your question, I should remark that every artin algebra is noetherian (by Hopkins–Levitzki theorem), and therefore you can think of them as a particular class of noetherian rings/algebras which are closest to finite-dimensional algebras and can be studied by power tools and techniques of representation theory of algebras. But this is not a popular viewpoint towards artin algebras. They are usually considered as a generalization of finite-dimensional algebras, and are interesting/important objects of study on their own (not as an auxiliary structure for understanding noetherian rings/algebras).

I tried to provide a succinct answer to your question, but you can find a far better answer by yourself: As you proceed in your studies keep a sharp eye on the particular properties of artin algebras you use and try to imagine how hard ``life'' would be without them!