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.
of left (or right) ideals has a largest element; that is, there exists an n such that:
Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively right-ideal) is finitely generated.
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.
There are rings that are left-Noetherian and not right-Noetherian, and vice versa.
The following condition is also an equivalent condition for a ring R to be left-Noetherian and it is Hilbert's original formulation:[2] For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.
[3] 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.
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 Q2 isomorphic 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.
[10] (In the special case when S is commutative, this is known as Eakin's theorem.)
A unique factorization domain is not necessarily a Noetherian ring.
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.
This is because there is a bijection between the left and right ideals of the group ring in this case, via the
Namely, given a ring R, the following are equivalent: The endomorphism ring of an indecomposable injective module is local[16] 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).