Noetherian ring

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 I 1 ⊆ I 2 ⊆ I 3 ⊆ ⋯ {\displaystyle I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq \cdots } of left (or right) ideals has a largest element; that is, there exists an n such that: I n = I n + 1 = ⋯ . {\displaystyle I_{n}=I_{n+1}=\cdots .}
Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian.
Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on the Noetherian property (for example, the Lasker–Noether theorem and the Krull intersection theorem).
Noetherian rings are named after Emmy Noether, but the importance of the concept was recognized earlier by David Hilbert, with the proof of Hilbert's basis theorem (which asserts that polynomial rings are Noetherian) and Hilbert's syzygy theorem.
For noncommutative rings, it is necessary to distinguish between three very similar concepts:
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.
There are other, equivalent, definitions for a ring R to be left-Noetherian:
Similar results hold for right-Noetherian rings.
The following condition is also an equivalent condition for a ring R to be left-Noetherian and it is Hilbert's original formulation:
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.)
Rings that are not Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings:
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,
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 Q isomorphic to Z, let R be the ring of homomorphisms f from Q 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(Q, Q), and S is finitely generated as a left R-module, but R is not left Noetherian.
A unique factorization domain is not necessarily a Noetherian ring. It does satisfy a weaker condition: the ascending chain condition on principal ideals. A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain.
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.
Consider the group ring R [ G ] {\displaystyle R[G]} of a group G {\displaystyle G} over a ring R {\displaystyle R} . It is a ring, and an associative algebra over R {\displaystyle R} if R {\displaystyle R} is commutative. For a group G {\displaystyle G} and a commutative ring R {\displaystyle R} , the following two conditions are equivalent.
This is because there is a bijection between the left and right ideals of the group ring in this case, via the R {\displaystyle R} -associative algebra homomorphism
Let G {\displaystyle G} be a group and R {\displaystyle R} a ring. If R [ G ] {\displaystyle R[G]} is left/right/two-sided Noetherian, then R {\displaystyle R} is left/right/two-sided Noetherian and G {\displaystyle G} is a Noetherian group. Conversely, if R {\displaystyle R} is a Noetherian commutative ring and G {\displaystyle G} is an extension of a Noetherian solvable group (i.e. a polycyclic group) by a finite group, then R [ G ] {\displaystyle R[G]} is two-sided Noetherian. On the other hand, however, there is a Noetherian group G {\displaystyle G} whose group ring over any Noetherian commutative ring is not two-sided Noetherian.
Many important theorems in ring theory (especially the theory of commutative rings) rely on the assumptions that the rings are Noetherian.
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:
The endomorphism ring of an indecomposable injective module is local and thus Azumaya's theorem says that, over a left Noetherian ring, each indecomposable decomposition of an injective module is equivalent to one another (a variant of the Krull–Schmidt theorem).