Noetherian ring

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:

${\displaystyle I_{1}\subseteq \cdots \subseteq I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq \cdots }$

there exists an n such that:

${\displaystyle I_{n}=I_{n+1}=\cdots .}$

There are other equivalent formulations of the definition of a Noetherian ring and these are outlined later in the article.

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 the 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:

• 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:

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. (The result is due to I. S. Cohen.)

Properties

• Z is a Noetherian ring, a fact which is exploited in the usual proof that every non-unit integer is divisible by at least one prime, although it's usually stated as "every non-empty set of integers has a minimal element with respect to divisibility".
• If R is a Noetherian ring, then R[X] is Noetherian by the Hilbert basis theorem. By induction, R[X1, ..., Xn] is a Noetherian ring. Also, 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

Rings that are not Noetherian tend to be (in some sense) very large. Here are three examples of non-Noetherian rings:

• The ring of polynomials in infinitely-many variables, X1, X2, X3, etc. The sequence of ideals (X1), (X1, X2), (X1, X2, X3), 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), (21/2), (21/4), (21/8), ...
• The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let In be the ideal of all continuous functions f such that f(x) = 0 for all xn. The sequence of ideals I0, I1, I2, etc., is an ascending chain that does not terminate.

However, a non-Noetherian ring can be a subring of a Noetherian ring: trivially because any integral domain is a subring of a field. For a less trivial,

• The ring of rational functions generated by x and y/xn over a field k is a subring of the field k(x,y) in only two variables.

Indeed, there are rings that are left Noetherian, but not right Noetherian, so that one must be careful in measuring the "size" of a ring this way.

A unique factorization domain is not necessarily a noetherian ring. It does satisfy a weaker condition: the ascending chain condition on principal ideals.

Primary decomposition

{{#invoke:main|main}}

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 pi, and positive integers ei, with ${\displaystyle n=\prod _{i}{p_{i}}^{e_{i}}}$. In this case, the ideal (n) may be written as the intersection of the ideals (piei); that is, ${\displaystyle (n)=\cap _{i}({p_{i}}^{e_{i}})}$. This is referred to as a 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 xyQ, either xQ or ynQ for some positive integer n. In Z, the primary ideals are precisely the ideals of the form (pe) where 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 ${\displaystyle n=\prod _{i}{p_{i}}^{e_{i}}}$ for pi prime and ei positive, a primary decomposition of (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. Let R be a commutative Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals; that is:

${\displaystyle I=\bigcap _{i=1}^{t}Q_{i}}$

with Qi primary for all i and Rad(Qi) ≠ Rad(Qj) for ij. Furthermore, if:

${\displaystyle I=\bigcap _{i=1}^{k}P_{i}}$

is decomposition of I with Rad(Pi) ≠ Rad(Pj) for ij, and both decompositions of I are irredundant (meaning that no proper subset of either {Q1, ..., Qt} or {P1, ..., Pk} yields an intersection equal to I), t = k and (after possibly renumbering the Qi) Rad(Qi) = Rad(Pi) for all i.

For any primary decomposition of I, the set of all radicals, that is, the set {Rad(Q1), ..., Rad(Qt)} 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.