Artinian Ring

Definition

An Artinian ring is a ring in which every descending chain of ideals eventually stabilizes. This means that if you keep taking ideals in a sequence where each one is contained in the previous one, you will eventually reach a point where you can't go any further. This property connects closely with other important features like the structure of modules over the ring, and it plays a significant role in understanding both finite-dimensional representations and various aspects of homological algebra.

5 Must Know Facts For Your Next Test

1. An Artinian ring is characterized by the descending chain condition on ideals, which implies that every ideal can be expressed as a finite sum of simple ideals.
2. Artinian rings have a structure theorem which states they can be decomposed into direct sums of local Artinian rings.
3. Every field is an Artinian ring since it has no proper ideals and thus satisfies the descending chain condition vacuously.
4. The Jacobson radical of an Artinian ring is nilpotent, which means that some power of the radical will equal zero.
5. Artinian rings are closely linked to finite-dimensional representations of algebras over fields, where the artin condition implies finite length modules.

Review Questions

• How does the descending chain condition on ideals define an Artinian ring, and what implications does this have for its structure?
  • The descending chain condition on ideals in an Artinian ring indicates that any sequence of ideals where each ideal contains the next must stabilize after a finite number of steps. This leads to significant implications for its structure, as it allows for a decomposition into simple components. Essentially, Artinian rings can be expressed as direct sums of local Artinian rings, reflecting their tightly controlled structure and simplifying the analysis of their modules.
• Compare and contrast Artinian rings with Noetherian rings in terms of their definitions and properties.
  • While Artinian rings are defined by the descending chain condition on ideals, Noetherian rings are defined by the ascending chain condition. This difference leads to distinct properties: for instance, every ideal in a Noetherian ring can be generated by finitely many elements, whereas in an Artinian ring, every ideal can be expressed as a finite sum of simple ideals. The interplay between these two types of rings reveals essential aspects of their respective module theories and how they relate to algebraic structures.
• Evaluate how the characteristics of an Artinian ring influence its modules and their representation theory.
  • The characteristics of an Artinian ring significantly influence its modules by ensuring that all modules over it have finite length. This means every module can be broken down into simple submodules, leading to a clear structure in representation theory. As a result, one can analyze representations using simple modules as building blocks, facilitating easier classification and understanding of representations over Artinian rings. Additionally, the finiteness conditions yield rich results in homological algebra, such as projective resolutions being manageable within this framework.

Related terms

Noetherian Ring:

A Noetherian ring is a ring in which every ascending chain of ideals stabilizes, meaning that if you keep adding ideals, you will eventually reach a point where you can't add anymore.

Semisimple Module: A semisimple module is a module that is a direct sum of simple modules, which are modules that have no submodules other than zero and themselves.

Artinian Module: An Artinian module is a module that satisfies the descending chain condition on submodules, similar to the definition of an Artinian ring but in the context of modules.