Noetherian ring

In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. It is named after Emmy Noether.

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.

For commutative rings, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa.

Characterizations of Noetherian rings

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

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

Similar results hold for right-Noetherian rings.

It is also known that for a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.

Uses of Noetherian rings

The Noetherian property is central in ring theory and in areas that make heavy use of rings, such as algebraic geometry. The reason behind this is that the Noetherian property is in some sense the ring-theoretic analogue of finiteness. 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.

As another application, we mention Krull's principal ideal theorem: Every principal ideal in a commutative Noetherian ring has height one. This early result was the first to suggest that Noetherian rings possessed a deep theory of dimension.

Examples

• The integers.
• Any field, including fields of rational numbers, real numbers, and complex numbers. (A field only has two ideals - itself and (0).)
• The ring of polynomials in finitely-many variables over the integers or a field.

Rings that are not Noetherian tend to be (in some sense) very large. Here are two examples of non-Noetherian rings:

• 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.

Properties

• If R is a Noetherian ring, then R[X] is Noetherian by the Hilbert basis theorem. Also, R[[X]], the power series ring is a Noetherian ring.
• If R is a Noetherian ring and I is a two-sided ideal, then the factor ring R/I is also Noetherian.
• Every finitely-generated commutative algebra over a field is Noetherian. (This follows from the two previous properties.)
• Every left (right) Artinian ring is left (right) Noetherian. This is the Akizuki-Hopkins-Levitzki Theorem.
• A ring R is left-Noetherian if and only if every finitely generated left R-module is a Noetherian module.

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