In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any increasing sequence of left (or right) ideals:

there exists a natural number n such that:

Noetherian rings are named after Emmy Noether.

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem, the Krull intersection theorem, and Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.

Characterizations

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:[2]

For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.[3]

Properties

Examples

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 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 IR 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.[10] (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 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.

Key theorems

Many important theorems in ring theory (especially the theory of commutative rings) rely on the assumptions that the rings are Noetherian.

Commutative case

Non-commutative case

This section needs expansion. You can help by adding to it. (December 2019)

Implication on injective modules

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[15] 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).

See also

Notes

  1. ^ a b Lam (2001), p. 19
  2. ^ Eisenbud 1995, Exercise 1.1.
  3. ^ Cohen, Irvin S. (1950). "Commutative rings with restricted minimum condition". Duke Mathematical Journal. 17 (1): 27–42. doi:10.1215/S0012-7094-50-01704-2. ISSN 0012-7094.
  4. ^ Matsumura, Theorem 3.5.
  5. ^ Matsumura, Theorem 3.6.
  6. ^ a b Anderson & Fuller 1992, Proposition 18.13.
  7. ^ Bourbaki 1989, Ch III, §2, no. 10, Remarks at the end of the number
  8. ^ Hotta, Takeuchi & Tanisaki (2008, §D.1, Proposition 1.4.6)
  9. ^ The ring of stable homotopy groups of spheres is not noetherian
  10. ^ Formanek & Jategaonkar 1974, Theorem 3
  11. ^ Eisenbud, Proposition 3.11.
  12. ^ Anderson & Fuller 1992, Theorem 25.6. (b)
  13. ^ Anderson & Fuller 1992, Theorem 25.8.
  14. ^ Anderson & Fuller 1992, Corollary 26.3.
  15. ^ Anderson & Fuller 1992, Lemma 25.4.

References