Noetherian ring

Last updated

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:

Contents

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.

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 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.

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] 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.)

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.

Noetherian group rings

Consider the group ring of a group over a ring . It is a ring, and an associative algebra over if is commutative. For a group and a commutative ring , 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 -associative algebra homomorphism

Let be a group and a ring. If is left/right/two-sided Noetherian, then is left/right/two-sided Noetherian and is a Noetherian group. Conversely, if is a Noetherian commutative ring and is an extension of a Noetherian solvable group (i.e. a polycyclic group) by a finite group, then is two-sided Noetherian. On the other hand, however, there is a Noetherian group whose group ring over any Noetherian commutative ring is not two-sided Noetherian. [11] :423,Theorem 38.1

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

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 [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).

See also

Notes

  1. 1 2 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 1989 , Theorem 3.5.
  5. Matsumura 1989 , Theorem 3.6.
  6. 1 2 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. Ol’shanskiĭ, Aleksandr Yur’evich (1991). Geometry of defining relations in groups. Mathematics and Its Applications. Soviet Series. Vol. 70. Translated by Bakhturin, Yu. A. Dordrecht: Kluwer Academic Publishers. doi:10.1007/978-94-011-3618-1. ISBN   978-0-7923-1394-6. ISSN   0169-6378. MR   1191619. Zbl   0732.20019.
  12. Eisenbud 1995 , Proposition 3.11.
  13. Anderson & Fuller 1992 , Theorem 25.6. (b)
  14. Anderson & Fuller 1992 , Theorem 25.8.
  15. Anderson & Fuller 1992 , Corollary 26.3.
  16. Anderson & Fuller 1992 , Lemma 25.4.

Related Research Articles

<span class="mw-page-title-main">Integral domain</span> Commutative ring with no zero divisors other than zero

In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

<span class="mw-page-title-main">Ring (mathematics)</span> Algebraic structure with addition and multiplication

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, a unique factorization domain (UFD) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain in which every non-zero non-unit element can be written as a product of irreducible elements, uniquely up to order and units.

<span class="mw-page-title-main">Commutative ring</span> Algebraic structure

In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings.

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

<span class="mw-page-title-main">Commutative algebra</span> Branch of algebra that studies commutative rings

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers ; and p-adic integers.

<span class="mw-page-title-main">Ring theory</span> Branch of algebra

In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type.

In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules over a ring, by keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below.

In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if Q is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module Y, any module homomorphism from this submodule to Q can be extended to a homomorphism from all of Y to Q. This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in some detail in the textbook.

In mathematics, specifically abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.

In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact.

In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways.

In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's lemma applies to them. In algebraic geometry, a completion of a ring of functions R on a space X concentrates on a formal neighborhood of a point of X: heuristically, this is a neighborhood so small that all Taylor series centered at the point are convergent. An algebraic completion is constructed in a manner analogous to completion of a metric space with Cauchy sequences, and agrees with it in the case when R has a metric given by a non-Archimedean absolute value.

In commutative algebra, an element b of a commutative ring B is said to be integral overA, a subring of B, if there are n ≥ 1 and aj in A such that

In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.

In commutative algebra, an integrally closed domainA is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A which is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed, as shown by the following chain of class inclusions:

This is a glossary of commutative algebra.

References