🧮Commutative Algebra Unit 7 – Noetherian and Artinian Rings

Key Concepts and Definitions

• Noetherian rings are rings that satisfy the ascending chain condition (ACC) on ideals
  • ACC states that every ascending chain of ideals $I_1 \subseteq I_2 \subseteq \cdots$ eventually stabilizes, meaning there exists an $n$ such that $I_n = I_{n+1} = \cdots$
• Artinian rings are rings that satisfy the descending chain condition (DCC) on ideals
  • DCC states that every descending chain of ideals $I_1 \supseteq I_2 \supseteq \cdots$ eventually stabilizes, meaning there exists an $n$ such that $I_n = I_{n+1} = \cdots$
• An ideal $I$ in a ring $R$ is finitely generated if there exist elements $a_1, \ldots, a_n \in R$ such that $I = (a_1, \ldots, a_n) = \{r_1a_1 + \cdots + r_na_n \mid r_1, \ldots, r_n \in R\}$
• A maximal ideal is a proper ideal $M$ of a ring $R$ such that there are no ideals strictly between $M$ and $R$
• A prime ideal is a proper ideal $P$ of a ring $R$ such that for any two elements $a, b \in R$, if $ab \in P$, then either $a \in P$ or $b \in P$
• The Krull dimension of a ring $R$ is the supremum of the lengths of all chains of prime ideals in $R$
• The Hilbert Basis Theorem states that if $R$ is a Noetherian ring, then the polynomial ring $R[x]$ is also Noetherian

Historical Context and Development

• The study of Noetherian and Artinian rings emerged in the early 20th century as part of the development of abstract algebra and commutative algebra
• Emmy Noether, a German mathematician, made significant contributions to the theory of rings and introduced the concept of Noetherian rings in the 1920s
  • Noether's work on chain conditions and finiteness properties of rings laid the foundation for the study of Noetherian and Artinian rings
• The term "Artinian" was introduced later, named after Emil Artin, an Austrian mathematician who made important contributions to algebra and number theory
• The development of Noetherian and Artinian rings was motivated by the need to understand the structure and properties of rings in a more abstract and general setting
• The study of these rings has led to important results and applications in various areas of mathematics, including algebraic geometry, number theory, and representation theory
• Noetherian and Artinian rings have become fundamental concepts in commutative algebra and are widely studied and applied in modern mathematics

Properties of Noetherian Rings

• Every ideal in a Noetherian ring is finitely generated
  • This property is equivalent to the ascending chain condition on ideals
• Every subring and quotient ring of a Noetherian ring is also Noetherian
• The polynomial ring $R[x]$ over a Noetherian ring $R$ is Noetherian (Hilbert Basis Theorem)
• Noetherian rings have a finite number of minimal prime ideals
• In a Noetherian ring, every non-empty set of ideals has a maximal element with respect to inclusion
• Noetherian rings satisfy the maximum condition on ideals, meaning every non-empty set of ideals has a maximal element
• The prime ideals in a Noetherian ring satisfy the descending chain condition
• Noetherian rings have finite Krull dimension

Properties of Artinian Rings

• Every ideal in an Artinian ring is finitely generated
  • This property is equivalent to the descending chain condition on ideals
• Every quotient ring of an Artinian ring is also Artinian
• Artinian rings have a finite number of maximal ideals
  • In fact, an Artinian ring has only finitely many prime ideals
• In an Artinian ring, every non-empty set of ideals has a minimal element with respect to inclusion
• Artinian rings satisfy the minimum condition on ideals, meaning every non-empty set of ideals has a minimal element
• The Krull dimension of an Artinian ring is always zero
• Every Artinian ring is Noetherian, but the converse is not true in general
• Artinian rings are a special case of Noetherian rings with additional finiteness properties

Relationships and Comparisons

• Every Artinian ring is Noetherian, but not every Noetherian ring is Artinian
  • For example, the ring of integers $\mathbb{Z}$ is Noetherian but not Artinian
• The concepts of Noetherian and Artinian rings are dual to each other in the sense that they impose chain conditions on ideals in opposite directions (ascending vs. descending)
• Fields and principal ideal domains (PIDs) are both Noetherian and Artinian
• If a ring $R$ is Noetherian and has Krull dimension zero, then $R$ is Artinian
• In a Noetherian ring, the concepts of maximal ideals and prime ideals coincide for ideals of height zero
• Noetherian rings can have infinite Krull dimension, while Artinian rings always have Krull dimension zero
• The properties of Noetherian and Artinian rings are preserved under certain ring constructions, such as taking subrings, quotient rings, and polynomial rings (in the Noetherian case)

Examples and Counterexamples

• The ring of integers $\mathbb{Z}$ is Noetherian but not Artinian
  • Every ideal in $\mathbb{Z}$ is finitely generated (principal), but there are infinite descending chains of ideals, such as $(2) \supseteq (4) \supseteq (8) \supseteq \cdots$
• Fields ($\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$) and finite rings ($\mathbb{Z}/n\mathbb{Z}$) are both Noetherian and Artinian
• The polynomial ring $k[x]$ over a field $k$ is Noetherian (by Hilbert Basis Theorem) but not Artinian
  • The ideals $(x) \supseteq (x^2) \supseteq (x^3) \supseteq \cdots$ form an infinite descending chain
• The ring of continuous functions on the real line, $C(\mathbb{R})$, is neither Noetherian nor Artinian
• The ring of formal power series $k[[x]]$ over a field $k$ is Noetherian but not Artinian
• The ring of algebraic integers is Noetherian but not Artinian
• The ring of rational functions $k(x)$ over a field $k$ is Artinian but not Noetherian

Theorems and Proofs

• Hilbert Basis Theorem: If $R$ is a Noetherian ring, then the polynomial ring $R[x]$ is also Noetherian
  • Proof idea: Use induction on the number of variables and the fact that ideals in $R[x]$ are finitely generated when $R$ is Noetherian
• Krull Intersection Theorem: In a Noetherian ring $R$, if $I$ is an ideal and $a \in R$, then $\bigcap_{n=1}^\infty (I^n + (a)) = I(\bigcap_{n=1}^\infty (I^n + (a)))$
  • This theorem has important consequences in commutative algebra and algebraic geometry
• Artin-Rees Lemma: Let $R$ be a Noetherian ring, $I$ an ideal, and $M$ a finitely generated $R$-module. For any submodule $N$ of $M$, there exists a positive integer $k$ such that $I^nM \cap N \subseteq I^{n-k}N$ for all $n \geq k$
  • This lemma is crucial in the study of completions and the structure of finitely generated modules over Noetherian rings
• Hopkins-Levitzki Theorem: A ring $R$ is Artinian if and only if it is Noetherian and has Krull dimension zero
  • Proof idea: Use the fact that in a Noetherian ring, prime ideals satisfy DCC, and in an Artinian ring, the Krull dimension is always zero

Applications in Algebra and Beyond

• Noetherian and Artinian rings play a fundamental role in the structure theory of commutative rings and modules
  • Many important results in commutative algebra rely on the properties of these rings
• In algebraic geometry, Noetherian rings are used to study the properties of algebraic varieties and schemes
  • The Hilbert Basis Theorem ensures that the coordinate rings of affine varieties are Noetherian
• Noetherian and Artinian rings are used in the study of dimension theory and the classification of commutative rings
• The concepts of Noetherian and Artinian rings have been generalized to non-commutative rings and modules, leading to important results in representation theory and non-commutative algebra
• Noetherian rings and modules appear in the study of invariant theory and the representation theory of finite groups
• The properties of Noetherian and Artinian rings are used in the study of homological algebra and the construction of derived functors
• Noetherian rings have applications in number theory, particularly in the study of algebraic number fields and their rings of integers