7.3 Artinian rings and the relationship with Noetherian rings

Artinian rings are a special class of rings with unique properties. They satisfy the descending chain condition on ideals, have finite length, and possess finitely many prime ideals. These characteristics set them apart from other ring structures.

Artinian rings are closely related to Noetherian rings, but with key differences. While all Artinian rings are Noetherian, the reverse isn't true. Understanding these distinctions helps in classifying and analyzing various ring structures in algebra.

Artinian Rings and Their Relationship with Noetherian Rings

Definition of Artinian rings

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• Ring R satisfies descending chain condition (DCC) on ideals any descending chain $I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots$ eventually stabilizes
• Integer n exists where $I_n = I_{n+1} = I_{n+2} = \cdots$
• Every non-empty set of ideals has a minimal element
• Every ideal finitely co-generated
• Finite rings, fields, and Artinian local rings exemplify Artinian rings

Artinian vs Noetherian rings

• Both satisfy chain conditions on ideals and exhibit finiteness conditions
• Artinian rings satisfy DCC, Noetherian rings satisfy ascending chain condition (ACC)
• Artinian implies Noetherian, not vice versa
• Artinian rings have finite length, Noetherian rings may have infinite length
• Artinian rings possess finitely many prime ideals, Noetherian rings may have infinitely many
• Artinian rings uniquely decompose into local Artinian rings, Noetherian rings lack this property

Equivalence of Artinian conditions

• Krull dimension supremum of lengths of chains of prime ideals
• Proof steps:

1. Demonstrate Artinian implies Noetherian
2. Establish Artinian rings have Krull dimension 0
3. Show Noetherian rings with Krull dimension 0 are Artinian

• Characterizes Artinian rings via Noetherian property and Krull dimension
• Links chain conditions to geometric ring properties

Identification of Artinian rings

• Verify descending chain condition on ideals
• Confirm every non-empty set of ideals has a minimal element
• Check if ring Noetherian with Krull dimension 0
• Analyze ideal structure, investigate prime spectrum, utilize known properties (PIDs, local rings)
• Examine quotient rings of polynomial rings, matrix rings over fields or Artinian rings, group rings of finite groups over fields

Examples of ring classifications

• Both Artinian and Noetherian finite rings, fields, $\mathbb{Z}/n\mathbb{Z}$ (n positive integer)
• Noetherian but not Artinian $\mathbb{Z}$ (integers), $k[x]$ (polynomial ring over field)
• Non-Noetherian polynomial ring in infinitely many variables
• Rings with infinite descending chains of ideals
• Examples highlight strictness of Artinian implies Noetherian implication
• Demonstrate Krull dimension's role in distinguishing Artinian and Noetherian rings