Krull dimension

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, 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:

$I_1\subseteq \cdots \subseteq I_{k-1}\subseteq I_k\subseteq I_{k+1}\subseteq \cdots$

there exists an n such that:

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

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

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:

• Every left ideal I in R is finitely generated, i.e. there exist elements a₁, ..., a_n in I such that I = Ra₁ + ... + Ra_n.
• Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion.

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.)

Hilbert's basis theorem

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If R is a ring, let R[X] denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is "not too large", in the sense that if R is Noetherian, the same must be true for R[X]. Formally,

Hilbert's Basis Theorem. If R is a Noetherian ring, then R[X] is a Noetherian ring.

Corollary. If R is a Noetherian ring, then R[X₁, ..., X_n] is a Noetherian ring.

For a proof of this result, see the corresponding section on the Hilbert's basis theorem page. Geometrically, the result asserts that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in n variables is generally a geometric object (such as a curve or a surface) in n-space).

Primary decomposition

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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_i, with $n=\prod _i{p_i}^{e_i}$. In this case, the ideal (n) may be written as the intersection of the ideals (p_i^e_i); that is, $(n)={\cap }_i({p_i}^{e_i})$. 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 xy ∈ Q, either x ∈ Q or yⁿ ∈ Q for some positive integer n. In Z, the primary ideals are precisely the ideals of the form (p^e) 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 $n=\prod _i{p_i}^{e_i}$ for p_i prime and e_i 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:

$I=\bigcap _{i=1}^t{Q}_i$

with Q_i primary for all i and Rad(Q_i) ≠ Rad(Q_j) for i ≠ j. Furthermore, if:

$I=\bigcap _{i=1}^k{P}_i$

is decomposition of I with Rad(P_i) ≠ Rad(P_j) for i ≠ j, and both decompositions of I are irredundant (meaning that no proper subset of either {Q₁, ..., Q_t} or {P₁, ..., P_k} yields an intersection equal to I), t = k and (after possibly renumbering the Q_i) Rad(Q_i) = Rad(P_i) for all i.

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

Uses

The Noetherian property is central in ring theory and in areas that make heavy use of rings, such as algebraic geometry. The reason behind this is that the Noetherian property is in some sense the ring-theoretic analogue of finiteness. For example, the fact that polynomial rings over a field are Noetherian allows one to prove that any infinite set of polynomial equations can be replaced with a finite set with the same solutions.

Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals. This early result was the first to suggest that Noetherian rings possessed a deep theory of dimension.

See also: Artin–Rees lemma.

Examples

• Any field, including fields of rational numbers, real numbers, and complex numbers. (A field only has two ideals — itself and (0).)
• Any principal ideal domain, such as the integers, is Noetherian since every ideal is generated by a single element.
• 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 $\mathfrak{g}$ is a both left and right noetherian ring; this follows from the fact that the associated graded ring of U is a quotient of $\mathrm{Sym}(\mathfrak{g})$, which is a polynomial ring over a field; thus, noetherian.
• 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 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), (2^1/2), (2^1/3), (2^1/4), ...
• The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I_n be the ideal of all continuous functions f such that f(x) = 0 for all x ≥ n. 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.

Properties

• If R is a Noetherian ring, then R[X] is Noetherian by the Hilbert basis theorem. 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.
• Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two previous properties.)
• 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 ring R is left-Noetherian if and only if every finitely generated left R-module is a Noetherian module.
• 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 commuative Noetherian ring, there are only finitely many minimal prime ideals.

See also

• Krull–Akizuki theorem
• Noetherian scheme
• Artinian ring

References

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• Nicolas Bourbaki, Commutative algebra
• Chapter X of Template:Lang Algebra
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