Artinian ring

Definition

An Artinian ring is a ring in which every descending chain of ideals eventually stabilizes. This property indicates that the ring has a certain 'finiteness' in its structure. Artinian rings are closely related to the concept of Noetherian rings, but while Noetherian rings focus on ascending chains of ideals, Artinian rings emphasize descending chains, making them important in the study of prime ideals and Krull dimension.

5 Must Know Facts For Your Next Test

1. In an Artinian ring, every non-empty collection of ideals has a minimal element under inclusion, highlighting the finite nature of its ideal structure.
2. Artinian rings can be characterized by their finite length as modules over themselves, meaning they can be decomposed into a finite number of simple modules.
3. Every Artinian ring is also Noetherian if it is also commutative, but not all Noetherian rings are Artinian.
4. The prime ideals of an Artinian ring are always finitely generated and correspond to simple modules over the ring.
5. A finite-dimensional algebra over a field is an example of an Artinian ring, showcasing how these rings appear in various algebraic structures.

Review Questions

• How do Artinian rings relate to the concept of descending chains of ideals, and why is this important?
  • Artinian rings are defined by their property that every descending chain of ideals stabilizes. This means that once you start listing ideals from largest to smallest, you will eventually reach a point where no new smaller ideals can be found. This stability is crucial as it leads to conclusions about the structure and behavior of the ring, particularly when examining its prime ideals and overall dimensional properties.
• Discuss the significance of maximal ideals in Artinian rings and how they relate to the structure of these rings.
  • Maximal ideals play a critical role in Artinian rings because they help define the ring's structure. In an Artinian ring, every maximal ideal is also prime, and since there are only finitely many maximal ideals, this gives insight into the overall organization of the ring. The existence of these maximal ideals helps determine the simple modules over the ring and further illustrates its finite length property.
• Evaluate the implications of a ring being both Noetherian and Artinian. What does this duality tell us about its prime ideals?
  • When a ring is both Noetherian and Artinian, it possesses a unique structure characterized by its finite dimensionality and compactness in terms of its prime ideals. This duality means that every ideal can be represented as being generated by a finite number of elements, and all chains of primes stabilize. Consequently, all prime ideals must also be maximal since they cannot extend indefinitely without becoming non-finite. This unique configuration allows for simpler analysis and classification of both modules and representations associated with the ring.

Related terms

Noetherian ring: A ring is Noetherian if every ascending chain of ideals stabilizes, meaning it has a finite number of generators for every ideal.

Radical of an ideal: The radical of an ideal is the set of elements in a ring that raise to some power belong to the ideal, which helps in analyzing properties of prime ideals.

Maximal ideal: A maximal ideal is an ideal that is not contained in any other proper ideal, playing a key role in the structure of both Artinian and Noetherian rings.