Noetherian ring

In abstract algebra, a Noetherian ring, named after Emmy Noether, is a ring that satisfies the ascending chain condition on ideals. Explicitly this means: given an increasing sequence of ideals

there exists an n for which

Since a principal ideal domain is a ring where each ideal can be generated by one element, a Noetherian ring can be considered as its generalization. Examples of Noetherian rings are number rings, polynomial rings over fields, the ring of formal power series, and group rings of polycyclic groups over fields.

Introduction

Polynomial rings 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

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, partially ordered by inclusion, 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. (The result is due to I. S. Cohen.)

Uses

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 fact that polynomial rings over a field are Noetherian allows one to prove that any infinite set of polynomial equations can be replaced with a finite set with the same solutions.

Krull's principal ideal theorem is an important property of Noetherian rings. It states that every principal ideal in a commutative Noetherian ring has height one; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals. This early result was the first to suggest that Noetherian rings possessed a deep theory of dimension.

Examples

• Any field, including fields of rational numbers, real numbers, and complex numbers. (A field only has two ideals — itself and (0).)
• Any principal ideal domain, such as the integers.
• 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 is not Noetherian: 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.

However, a non-Noetherian ring can be a subring of a Noetherian ring:

• The ring of rational functions generated by x and y/xⁿ over a field k is a subring of the field k(x,y) in only two variables.

Indeed, there are rings that are left noetherian, but not right noetherian, so that one must be careful in measuring the "size" of ring this way.

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.
• A left-Noetherian ring is left coherent and a Noetherian non-commutative integral domain is an Ore domain.

References

• Chapter X of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0-201-55540-0