In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that:
Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian.
The following condition is also an equivalent condition for a ring R to be left-Noetherian and it is Hilbert's original formulation:
Given a sequence of elements in R, there exists an integer such that each is a finite linear combination with coefficients in R.
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. However, it is not enough to ask that all the maximal ideals are finitely generated, as there is a non-Noetherian local ring whose maximal ideal is principal (see a counterexample to Krull’s intersection theorem at Local ring#Commutative case.)
If a commutative ring admits a faithful Noetherian module over it, then the ring is a Noetherian ring.
(Eakin–Nagata) If a ring A is a subring of a commutative Noetherian ring B such that B is a finitely generated module over A, then A is a Noetherian ring.
Similarly, if a ring A is a subring of a commutative Noetherian ring B such that B is faithfully flat over A (or more generally exhibits A as a pure subring), then A is a Noetherian ring (see the "faithfully flat" article for the reasoning).
Every localization of a commutative Noetherian ring is Noetherian.
A consequence of the Akizuki–Hopkins–Levitzki theorem is that every left Artinian ring is left Noetherian. Another consequence is that a left Artinian ring is right Noetherian if and only if it is right Artinian. The analogous statements with "right" and "left" interchanged are also true.
The enveloping algebra U of a finite-dimensional Lie algebra is a both left and right Noetherian ring; this follows from the fact that the associated graded ring of U is a quotient of , which is a polynomial ring over a field; thus, Noetherian. For the same reason, the Weyl algebra, and more general rings of differential operators, are Noetherian.
The ring of polynomials in finitely-many variables over the integers or a field is Noetherian.
Rings that are not Noetherian tend to be (in some sense) very large. Here are some 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 all algebraic integers is not Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (21/2), (21/4), (21/8), ...
The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let In 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. Since any integral domain is a subring of a field, any integral domain that is not Noetherian provides an example. To give a less trivial example,
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 right Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way. For example, if L is a subgroup of Q2isomorphic to Z, let R be the ring of homomorphisms f from Q2 to itself satisfying f(L) ⊂ L. Choosing a basis, we can describe the same ring R as
This ring is right Noetherian, but not left Noetherian; the subset I ⊂ R consisting of elements with a = 0 and γ = 0 is a left ideal that is not finitely generated as a left R-module.
If R is a commutative subring of a left Noetherian ring S, and S is finitely generated as a left R-module, then R is Noetherian. (In the special case when S is commutative, this is known as Eakin's theorem.) However this is not true if R is not commutative: the ring R of the previous paragraph is a subring of the left Noetherian ring S = Hom(Q2, Q2), and S is finitely generated as a left R-module, but R is not left Noetherian.
A valuation ring is not Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in algebraic geometry but is not Noetherian.
Many important theorems in ring theory (especially the theory of commutative rings) rely on the assumptions that the rings are Noetherian.
Over a commutative Noetherian ring, each ideal has a primary decomposition, meaning that it can be written as an intersection of finitely many primary ideals (whose radicals are all distinct) where an ideal Q is called primary if it is proper and whenever xy ∈ Q, either x ∈ Q or yn ∈ Q for some positive integer n. For example, if an element is a product of powers of distinct prime elements, then and thus the primary decomposition is a direct generalization of prime factorization of integers and polynomials.
A Noetherian ring is defined in terms of ascending chains of ideals. The Artin–Rees lemma, on the other hand, gives some information about a descending chain of ideals given by powers of ideals . It is a technical tool that is used to prove other key theorems such as the Krull intersection theorem.
The dimension theory of commutative rings behaves poorly over non-Noetherian rings; the very fundamental theorem, Krull's principal ideal theorem, already relies on the "Noetherian" assumption. Here, in fact, the "Noetherian" assumption is often not enough and (Noetherian) universally catenary rings, those satisfying a certain dimension-theoretic assumption, are often used instead. Noetherian rings appearing in applications are mostly universally catenary.
Given a ring, there is a close connection between the behaviors of injective modules over the ring and whether the ring is a Noetherian ring or not. Namely, given a ring R, the following are equivalent:
R is a left Noetherian ring.
(Bass) Each direct sum of injective left R-modules is injective.
Anderson, Frank W.; Fuller, Kent R. (1992), Rings and categories of modules, Graduate Texts in Mathematics, vol. 13 (2 ed.), New York: Springer-Verlag, pp. x+376, doi:10.1007/978-1-4612-4418-9, ISBN 0-387-97845-3, MR 1245487
Atiyah, M. F., MacDonald, I. G. (1969). Introduction to commutative algebra. Addison-Wesley-Longman. ISBN 978-0-201-40751-8