In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. It is named after Emmy Noether. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... Amalie Emmy Noether [1] (March 23, 1882 – April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ...

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. In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... Amalie Emmy Noether [1] (March 23, 1882 – April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ... In mathematics, a poset P is said to satisfy the ascending chain condition (ACC) if every ascending chain a1 ≤ a2 ≤ ... of elements of P is eventually stationary, that is, there is some positive integer n such that am = an for all m > n. ...

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. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

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. In set theory, a set is called non-empty (or nonempty) if it contains at least one element, and is therefore not the empty set. ... In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. The term minimal element is defined dually. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...

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. In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

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. In commutative algebra, Krulls principal ideal theorem, named after Wolfgang Krull (1899 - 1971), gives a bound on the height of a principal ideal in a Noetherian ring. ... In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R... In commutative algebra, the height of an ideal I in a ring R is the number of strict inclusions in the longest chain of prime ideals contained in I. In a Noetherian ring, Krulls height theorem says that the height of an ideal generated by n elements is no... In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. ...

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 integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

• 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.