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 ideals; that is, given any chain of ideals:

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.

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:

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

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

R=\left\{\left.\begin{bmatrix}a & \beta \\0 & \gamma \end{bmatrix} \, \right\vert\, a\in \mathbb{Z}, \beta\in \mathbb{Q},\gamma\in \mathbb{Q}\right\}.

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.[4] (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 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

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 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, (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 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:
I=\bigcap_{i=1}^t Q_i

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

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.

See also

References

External links

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