- #1
PsychonautQQ
- 784
- 10
Can somebody explain why this shouldn't surprise me? Apparently it's a corollary of the Akizuki–Hopkins–Levitzki theorem but the proof of that theorem isn't helping me really understand what's going on.
An Artinian ring is a type of ring in abstract algebra that has a finite length as a module over itself. This means that there is a finite chain of submodules that eventually reaches the entire ring. In other words, the ring can be broken down into a finite number of simple modules.
A Noetherian ring is a type of ring that satisfies the ascending chain condition for ideals. This means that for any ascending chain of ideals in the ring, there exists a finite number of steps before the chain stabilizes and becomes equal to a single ideal.
This makes sense because both Artinian and Noetherian rings have a finite structure, which means they have a limited number of submodules or ideals. Therefore, it is logical for an Artinian ring to also satisfy the ascending chain condition for ideals.
If an Artinian ring is also Noetherian, it means that the ring has both a finite length and satisfies the ascending chain condition for ideals. This has many important implications, such as simplifying the structure and properties of the ring, making it easier to study and understand.
No, not all Artinian rings are Noetherian. While the properties of finite length and satisfying the ascending chain condition for ideals are related, they are not equivalent. There are examples of Artinian rings that are not Noetherian, such as the Weyl algebra.