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; that is, given an increasing sequence of left 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:- 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.
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 in I such that.
- Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element.
The following condition is also an equivalent condition for a ring R to be left-Noetherian and it is Hilbert's original formulation:
- Given a sequence of elements in R, there exists an integer such that each is a finite linear combination with coefficients in R.
Properties
- If R is a Noetherian ring, then the polynomial ring is Noetherian by the Hilbert's basis theorem. By induction, is a Noetherian ring. Also,, the power series ring is a Noetherian ring.
- If is a Noetherian ring and is a two-sided ideal, then the quotient ring 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.
- A ring R is left-Noetherian if and only if every finitely generated left R-module is a Noetherian module.
- If a commutative ring admits a faithful Noetherian module over it, then the ring is a Noetherian ring.
- If a ring A is a subring of a commutative Noetherian ring B such that B is a finitely generated module over A, then A is a Noetherian ring.
- Similarly, if a ring A is a subring of a commutative Noetherian ring B such that B is faithfully flat over A, then A is a Noetherian ring.
- 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 Noetherian if and only if every direct sum of injective modules is injective. Every left injective module over a left Noetherian module can be decomposed as a direct sum of indecomposable injective modules.
- In a commutative Noetherian ring, there are only finitely many minimal prime ideals. Also, the descending chain condition holds on 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
- Any field, including fields of rational numbers, real numbers, and complex numbers, is Noetherian.
- 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 is a Noetherian domain in which every ideal is generated by at most two elements.
- 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. For the same reason, the Weyl algebra, and more general rings of differential operators, are Noetherian.
- The ring of polynomials in finitely-many variables over the integers or a field.
- The ring of polynomials in infinitely-many variables, X1, X2, X3, etc. The sequence of ideals,,, etc. is ascending, and does not terminate.
- The ring of all algebraic integers is not Noetherian. For example, it contains the infinite ascending chain of principal ideals:,,,,...
- 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 = 0 for all x ≥ n. The sequence of ideals I0, I1, I2, etc., is an ascending chain that does not terminate.
- The ring of stable homotopy groups of spheres is not Noetherian.
- The ring of rational functions generated by x and y/xn over a field k is a subring of the field k in only two variables.
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. 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, 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 rely on the assumptions that the rings are Noetherian.Commutative case
- Over a commutative Noetherian ring, each ideal has a primary decomposition, meaning that it can be written as an intersection of finitely many primary ideals where an ideal Q is called primary if it is proper and whenever xy ∈ Q, either x ∈ Q or yn ∈ Q for some positive integer n. For example, if an element is a product of powers of distinct prime elements, then and thus the primary decomposition is a direct generalization of prime factorization of integers and polynomials.
- A Noetherian ring is defined in terms of ascending chains of ideals. The Artin–Rees lemma, on the other hand, gives some information about a descending chain of ideals given by powers of ideals. It is a technical tool that is used to prove other key theorems such as the Krull intersection theorem.
- The dimension theory of commutative rings behaves poorly over non-Noetherian rings; the very fundamental theorem, Krull's principal ideal theorem, already relies on the "Noetherian" assumption. Here, in fact, the "Noetherian" assumption is often not enough and universally catenary rings, those satisfying a certain dimension-theoretic assumption, are often used instead. Noetherian rings appearing in applications are mostly universally catenary.
Non-commutative case
- Goldie's theorem
Implication on injective modules
- R is a left Noetherian ring.
- Each direct sum of injective left R-modules is injective.
- Each injective left R-module is a direct sum of indecomposable injective modules.
- There exists a cardinal number such that each injective left module over R is a direct sum of -generated modules.
- There exists a left R-module H such that every left R-module embeds into a direct sum of copies of H.