Table of contents

1 Introduction 2 Characterizations of Noetherian rings 3 Uses in algebraic geometry 4 Examples 5 Properties

Introduction

Rings of polynomials over fields have many special properties; properties that follow from the fact that polynomial rings are not, in some sense, "too large". Emmy Noether first discovered that the key property of polynomial rings is the ascending chain condition on ideals. Noetherian rings are named after her.

For noncommutative rings, we must distinguish between three very similar concepts:

• A ring is left Noetherian if it satisfies the ascending chain condition on left ideals.
• A ring is right Noetherian if it satisfies the ascending chain condition on right ideals.
• A ring is Noetherian if it is both left and right Noetherian.

Characterizations of Noetherian rings

There are other, equivalent, definitions for a ring R to be left Noetherian:

• Every ideal I in R is finitely generated, i.e. there exists elements a₁,...,a_n in I such that I = Ra₁ + ... + Ra_n.
• Any non-empty set of left ideals of R has a maximal element with respect to set inclusion.

Uses in algebraic geometry

The Noetherian property is central in algebraic geometry. For example, the Noetherian-ness of polynomial rings over a field allows us to prove that any infinite set of polynomial equations can be replaced with a finite set with the same solutions.

The Noetherian property is also important in the dimension theory of algebraic varieties. Suppose we take the solutions of a set of polynomial equations, and suppose the solution set is a variety of dimension n. Then by the Noetherian property if we add additional equations, we must eventually force the dimension to go down.

Examples

• The integers.
• Any field, including fields of rational numbers, real numbers, and complex numbers.
• The ring of polynomials in finitely-manys over the integers or a field.

• The ring of polynomials in infinitely-many variables, X₁, X₂, X₃, etc. The sequence of ideals (X₁), (X₁,X₂), (X₁,X₂, X₃), etc. is ascending, and does not terminate.
• The ring of continuous functions from the real numbers to the real numbers. Let I_n be the ideal of all continuous functions f such that f(x) = 0 for all x ≤ n. The sequence of ideals I₀, I₁, I₂, etc., is an ascending chain that does not terminate.

• Every field is Noetherian. (A field only has two ideals - itself and (0).)
• If R is a Noetherian ring, then R[X] is also Noetherian, by the Hilbert basis theorem.
• If R is a Noetherian and I is a two-sided ideal, then the factor ring R/I is also Noetherian.
• Every finitely-generated algebra over a field is Noetherian. (This follows from the two previous properties.)
• Every left Artinian ring is left Noetherian. This is the Hopkins-Levitzki Theorem.