# Noetherian ring

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, which means there is no infinite ascending sequence of left (or right) ideals; that is, given any chain of left (or right) ideals,

there exists an *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.

Algebraic structures |
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## Contents

## Characterizations[edit]

For noncommutative rings, it is necessary to distinguish between three very similar concepts:

- A ring is
**left-Noetherian**if it satisfies the ascending chain condition on left ideals. - A ring is
**right-Noetherian**if it satisfies the ascending chain condition on right ideals. - A ring is
**Noetherian**if it is both left- and right-Noetherian.

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:

- Every left ideal
*I*in*R*is finitely generated, i.e. there exist elements*a*_{1}, ...,*a*in_{n}*I*such that*I*=*Ra*_{1}+ ... +*Ra*_{n}.^{[1]} - Every non-empty set of left ideals of
*R*, partially ordered by inclusion, has a maximal element with respect to set inclusion.^{[1]}

Similar results hold for right-Noetherian rings.

For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated.^{[2]}

## Properties[edit]

- If
*R*is a Noetherian ring, then*R*[*X*] is Noetherian by the Hilbert basis theorem. By induction,*R*[*X*_{1}, ...,*X*] is a Noetherian ring. Also,_{n}*R*[[*X*]], the power series ring is a Noetherian ring. - If
*R*is a Noetherian ring and*I*is a two-sided ideal, then the factor ring*R*/*I*is also Noetherian. Stated differently, the image of any surjective ring homomorphism of a Noetherian ring is Noetherian. - Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.)
- A ring
*R*is left-Noetherian if and only if every finitely generated left*R*-module is a Noetherian module. - 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 right Artinian; the analogous statements with "right" and "left" interchanged are also true.
- A left Noetherian ring is left coherent and a left Noetherian domain is a left Ore domain.
- A ring is (left/right) Noetherian if and only if every direct sum of injective (left/right) modules is injective. Every injective module can be decomposed as direct sum of indecomposable injective modules.
- In a commutative Noetherian ring, there are only finitely many minimal prime ideals.
- In a commutative Noetherian domain
*R*, every element can be factorized into irreducible elements. Thus, if, in addition, irreducible elements are prime elements, then*R*is a unique factorization domain.

## Examples[edit]

- Any field, including fields of rational numbers, real numbers, and complex numbers, is Noetherian. (A field only has two ideals — itself and (0).)
- Any principal ideal ring, such as the integers, is Noetherian since every ideal is generated by a single element. This includes principal ideal domains and Euclidean domains.
- A Dedekind domain (e.g., rings of integers) is Noetherian since every ideal is generated by at most two elements. The "Noetherian" follows from the Krull–Akizuki theorem; the bounds on the number of the generators is a corollary of the Forster–Swan theorem (or basic ring theory).
- The coordinate ring of an affine variety is a Noetherian ring, as a consequence of the Hilbert basis theorem.
- 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.^{[3]}For the same reason, the Weyl algebra, and more general rings of differential operators, are Noetherian.^{[4]} - The ring of polynomials in finitely-many variables over the integers or a field.

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,
*X*_{1},*X*_{2},*X*_{3}, etc. The sequence of ideals (*X*_{1}), (*X*_{1},*X*_{2}), (*X*_{1},*X*_{2},*X*_{3}), etc. is ascending, and does not terminate. - The ring of algebraic integers is not Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (2
^{1/2}), (2^{1/4}), (2^{1/8}), ... - The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let
*I*be the ideal of all continuous functions_{n}*f*such that*f*(*x*) = 0 for all*x*≥*n*. The sequence of ideals*I*_{0},*I*_{1},*I*_{2}, etc., is an ascending chain that does not terminate. - The ring of stable homotopy groups of spheres is not Noetherian.
^{[5]}

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*/*x*^{n}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 **Q**^{2} isomorphic to **Z**, let *R* be the ring of homomorphisms *f* from **Q**^{2} 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.^{[6]} (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**^{2},**Q**^{2}), 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 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.

## Primary decomposition[edit]

In the ring **Z** of integers, an arbitrary ideal is of the form (*n*) for some integer *n* (where (*n*) denotes the set of all integer multiples of *n*). If *n* is non-zero, and is neither 1 nor −1, by the fundamental theorem of arithmetic, there exist primes *p _{i}*, and positive integers

*e*, with . In this case, the ideal (

_{i}*n*) may be written as the intersection of the ideals (

*p*); that is, . This is referred to as a

_{i}^{ei}*primary decomposition*of the ideal (

*n*).

In general, an ideal *Q* of a ring is said to be *primary* if *Q* is proper and whenever *xy* ∈ *Q*, either *x* ∈ *Q* or *y ^{n}* ∈

*Q*for some positive integer

*n*. In

**Z**, the primary ideals are precisely the ideals of the form (

*p*) where

^{e}*p*is prime and

*e*is a positive integer. Thus, a primary decomposition of (

*n*) corresponds to representing (

*n*) as the intersection of finitely many primary ideals.

Since the fundamental theorem of arithmetic applied to a non-zero integer *n* that is neither 1 nor −1 also asserts uniqueness of the representation for *p _{i}* prime and

*e*positive, a primary decomposition of (

_{i}*n*) is essentially

*unique*.

For all of the above reasons, the following theorem, referred to as the *Lasker–Noether theorem*, may be seen as a certain generalization of the fundamental theorem of arithmetic:

Lasker-Noether Theorem.LetRbe a commutative Noetherian ring and letIbe an ideal ofR. ThenImay be written as the intersection of finitely many primary ideals with distinct radicals; that is:

with

Qprimary for all_{i}iand Rad(Q) ≠ Rad(_{i}Q) for_{j}i≠j. Furthermore, if:

is decomposition of

Iwith Rad(P) ≠ Rad(_{i}P) for_{j}i≠j, and both decompositions ofIareirredundant(meaning that no proper subset of either {Q_{1}, ...,Q} or {_{t}P_{1}, ...,P} yields an intersection equal to_{k}I),t=kand (after possibly renumbering theQ) Rad(_{i}Q) = Rad(_{i}P) for all_{i}i.

For any primary decomposition of *I*, the set of all radicals, that is, the set {Rad(*Q*_{1}), ..., Rad(*Q _{t}*)} remains the same by the Lasker–Noether theorem. In fact, it turns out that (for a Noetherian ring) the set is precisely the assassinator of the module

*R*/

*I*; that is, the set of all annihilators of

*R*/

*I*(viewed as a module over

*R*) that are prime.

## See also[edit]

- Krull–Akizuki theorem
- Noetherian scheme
- Artinian ring
- Artin–Rees lemma
- Krull's principal ideal theorem

## References[edit]

- ^
^{a}^{b}Lam (2001), p. 19 **^**Cohen, I. 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.**^**Bourbaki 1989, Ch III, §2, no. 10, Remarks at the end of the number**^**Hotta, Takeuchi & Tanisaki (2008, §D.1, Proposition 1.4.6)**^**http://math.stackexchange.com/questions/1513353/the-ring-of-stable-homotopy-groups-of-spheres-is-not-noetherian**^**Formanek & Jategaonkar 1974, Theorem 3

- Nicolas Bourbaki, Commutative algebra
- Formanek, Edward; Jategaonkar, Arun Vinayak (1974). "Subrings of Noetherian rings".
*Proc. Amer. Math. Soc*.**46**(2): 181–186. doi:10.2307/2039890. - Hotta, Ryoshi; Takeuchi, Kiyoshi; Tanisaki, Toshiyuki (2008),
*D-modules, perverse sheaves, and representation theory*, Progress in Mathematics,**236**, Birkhäuser, doi:10.1007/978-0-8176-4523-6, ISBN 978-0-8176-4363-8, MR 2357361, Zbl 1292.00026 - Lam, T.Y. (2001).
*A first course in noncommutative rings*. New York: Springer. p. 19. ISBN 0387951830. - Chapter X of Lang, Serge (1993),
*Algebra*(Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001

## External links[edit]

- Hazewinkel, Michiel, ed. (2001) [1994], "Noetherian ring",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4