An Introduction to Noetherian Rings

What is a Noetherian Ring?

A ring $R$ is called Noetherian if it satisfies any of the following equivalent conditions:

1. Every ideal of $R$ is finitely generated
2. Every ascending chain of ideals $I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$ eventually stabilizes
3. Every non-empty set of ideals has a maximal element

This property, named after Emmy Noether, is fundamental in commutative algebra and algebraic geometry.

Examples

1. Every field is Noetherian (since its only ideals are $(0)$ and the whole ring)
2. The ring of integers $\mathbb{Z}$ is Noetherian
3. Any principal ideal domain (PID) is Noetherian
4. If $R$ is Noetherian, then the polynomial ring $R[x]$ is Noetherian (Hilbert’s Basis Theorem)

Non-Examples

1. The ring of all continuous functions $C[0,1]$ is not Noetherian
2. The ring of all polynomials in infinitely many variables over a field is not Noetherian

Why Are Noetherian Rings Important?

Noetherian rings are crucial because they provide finiteness conditions that make many algebraic problems tractable. For instance:

• In a Noetherian ring, every submodule of a finitely generated module is finitely generated
• The Lasker–Noether decomposition theorem states that in a Noetherian ring, every ideal has a primary decomposition
• In algebraic geometry, Noetherian rings ensure that schemes have nice geometric properties

The Ascending Chain Condition

The second characterization of Noetherian rings is particularly useful in proofs. If we have an ascending chain of ideals:

$I_1 \subseteq I_2 \subseteq I_3 \subseteq \cdots$

Then there exists some $n$ such that $I_n = I_{n+1} = I_{n+2} = \cdots$

This is called the ascending chain condition (ACC) and is often easier to verify than the other conditions.

A Key Theorem

One of the most important results about Noetherian rings is Hilbert’s Basis Theorem:

Theorem: If $R$ is a Noetherian ring, then $R[x]$ is also Noetherian.

This result extends inductively to show that $R[x_1,\ldots,x_n]$ is Noetherian for any finite number of variables.

Applications

Noetherian rings appear throughout mathematics:

1. In algebraic geometry, most rings of geometric origin are Noetherian
2. In commutative algebra, Noetherian hypotheses are essential for many structure theorems
3. In representation theory, Noetherian properties ensure finite-dimensional decompositions

Further Reading

For those interested in learning more about Noetherian rings, I recommend:

• Atiyah and Macdonald’s “Introduction to Commutative Algebra”
• Eisenbud’s “Commutative Algebra with a View Toward Algebraic Geometry”
• Reid’s “Undergraduate Commutative Algebra”