nLab noetherian ring

Contents

Contents

Idea

A Noetherian (or often, as below, noetherian) ring (or rng) is one where it is possible to do induction over its ideals, because the ordering of ideals by reverse inclusion is well-founded.

Definition

Noetherian rings

Every ring RR has a canonical RR-RR-bimodule structure, with left action α L:R×RR\alpha_L:R \times R \to R and right action α R:R×RR\alpha_R:R \times R \to R defined as the multiplicative binary operation on RR and biaction α:R×R×RR\alpha:R \times R \times R \to R defined as the ternary product on RR:

α L(a,b)ab\alpha_L(a, b) \coloneqq a \cdot b
α R(a,b)ab\alpha_R(a, b) \coloneqq a \cdot b
α(a,b,c)abc\alpha(a, b, c) \coloneqq a \cdot b \cdot c

Let TwoSidedIdeals(R)\mathrm{TwoSidedIdeals}(R) be the category of two-sided ideals in RR, whose objects are two-sided ideals II in RR, sub- R R - R R -bimodules of RR with respect to the canonical bimodule structure on RR, and whose morphisms are RR-RR-bimodule monomorphisms.

An ascending chain of two-sided ideals in RR is a direct sequence of two-sided ideals in RR, a sequence of two-sided ideals A:TwoSidedIdeals(R)A:\mathbb{N} \to \mathrm{TwoSidedIdeals}(R) with the following dependent sequence of RR-RR-bimodule monomorphisms: for natural number nn \in \mathbb{N}, a dependent RR-RR-bimodule monomorphism i n:A nA n+1i_n:A_{n} \hookrightarrow A_{n+1}.

A ring RR is Noetherian if it satisfies the ascending chain condition on its two-sided ideals: for every ascending chain of two-sided ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the RR-RR-bimodule monomorphism i n:A nA n+1i_n:A_{n} \hookrightarrow A_{n+1} is an RR-RR-bimodule isomorphism.

Left Noetherian rings

Let LeftIdeals(R)\mathrm{LeftIdeals}(R) be the category of left ideals in RR, whose objects are left ideals II in RR, sub-left-RR-modules of RR with respect to the canonical left module structure ()():R×RR(-)\cdot(-):R \times R \to R on RR, and whose morphisms are left RR-module monomorphisms.

An ascending chain of left ideals in RR is a direct sequence of left ideals in RR, a sequence of left ideals A:LeftIdeals(R)A:\mathbb{N} \to \mathrm{LeftIdeals}(R) with the following dependent sequence of left RR-module monomorphisms: for natural number nn \in \mathbb{N}, a dependent left RR-module monomorphism i n:A nA n+1i_n:A_{n} \hookrightarrow A_{n+1}.

A ring RR is left Noetherian if it satisfies the ascending chain condition on its left ideals: for every ascending chain of left ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the left RR-module monomorphism i n:A nA n+1i_n:A_{n} \hookrightarrow A_{n+1} is an left RR-module isomorphism.

Right Noetherian rings

Let RightIdeals(R)\mathrm{RightIdeals}(R) be the category of right ideals in RR, whose objects are right ideals II in RR, sub-right-RR-modules of RR with respect to the canonical right module structure ()():R×RR(-)\cdot(-):R \times R \to R on RR, and whose morphisms are right RR-module monomorphisms.

An ascending chain of right ideals in RR is a direct sequence of right ideals in RR, a sequence of right ideals A:RightIdeals(R)A:\mathbb{N} \to \mathrm{RightIdeals}(R) with the following dependent sequence of right RR-module monomorphisms: for natural number nn \in \mathbb{N}, a dependent right RR-module monomorphism i n:A nA n+1i_n:A_{n} \hookrightarrow A_{n+1}.

A ring RR is right Noetherian if it satisfies the ascending chain condition on its right ideals: for every ascending chain of right ideals (A,i n)(A, i_n) in RR, there exists a natural number mm \in \mathbb{N} such that for all natural numbers nmn \geq m, the right RR-module monomorphism i n:A nA n+1i_n:A_{n} \hookrightarrow A_{n+1} is an right RR-module isomorphism.

Examples

Example

Every field is a noetherian ring.

Example

Every principal ideal domain is a noetherian ring.

Example

For RR a Noetherian ring (e.g. a field by example ) then

  1. the polynomial algebra R[X 1,,X n]R[X_1, \cdots, X_n]

  2. the formal power series algebra R[[X 1,,X n]]R[ [ X_1, \cdots, X_n ] ]

over R in a finite number nn of coordinates are Noetherian.

Properties

Spectra of noetherian rings are glued together to define locally noetherian schemes.

General

One of the best-known properties is the Hilbert basis theorem. Let RR be a (unital) ring.

Theorem

(Hilbert) If RR is left Noetherian, then so is the polynomial algebra R[x]R[x]. (Similarly if “right” is substituted for “left”.)

Proof

(We adapt the proof from Wikipedia.) Suppose II is a left ideal of R[x]R[x] that is not finitely generated. Using the axiom of dependent choice, there is a sequence of polynomials f nIf_n \in I such that the left ideals I n(f 0,,f n1)I_n \coloneqq (f_0, \ldots, f_{n-1}) form a strictly increasing chain and f nII nf_n \in I \setminus I_n is chosen to have degree as small as possible. Putting d ndeg(f n)d_n \coloneqq \deg(f_n), we have d 0d 1d_0 \leq d_1 \leq \ldots. Let a na_n be the leading coefficient of f nf_n. The left ideal (a 0,a 1,)(a_0, a_1, \ldots) of RR is finitely generated; say (a 0,,a k1)(a_0, \ldots, a_{k-1}) generates. Thus we may write

(1)a k= i=0 k1r ia i a_k = \sum_{i=0}^{k-1} r_i a_i

The polynomial g= i=0 k1r ix d kd if ig = \sum_{i=0}^{k-1} r_i x^{d_k - d_i} f_i belongs to I kI_k, so f kgf_k - g belongs to II kI \setminus I_k. Also gg has degree d kd_k or less, and therefore so does f kgf_k - g. But notice that the coefficient of x d kx^{d_k} in f kgf_k - g is zero, by (1). So in fact f kgf_k - g has degree less than d kd_k, contradicting how f kf_k was chosen.

A homological characterization

Theorem

For a unital ring RR the following are equivalent:

  1. RR is left Noetherian
  2. Any small direct sum of injective left RR-modules is injective.
  3. Ext R k(A,)\operatorname{Ext}^k_R(A, \cdot) commutes with small direct sums for any finitely generated AA.

Direct sums here can be replaced by filtered colimits.

Proof

121 \Rightarrow 2: assume that RR is Noetherian and I αI_\alpha are injective modules. In order to verify that I:= αI αI := \bigoplus_\alpha I_\alpha is injective it is enough to show that for any ideal 𝔧\mathfrak{j} any morphism of left modules f:𝔧If : \mathfrak{j} \to I factors through 𝔧R\mathfrak{j} \to R. Since RR is Notherian, 𝔧\mathfrak{j} is finitely generated, so the image of ff lies in a finite sum I α 1I α nI_{\alpha_1} \oplus \dots \oplus I_{\alpha_n}. Thus an extension to RR exists by the injectivity of each I α kI_{\alpha_k}.

212 \Rightarrow 1: if RR is not left Noetherian then there is a sequence of left ideals 𝔧 1𝔧 2\mathfrak{j}_1 \subsetneq \mathfrak{j}_2 \subsetneq \dots. Take 𝔧:= k𝔧 k\mathfrak{j} := \bigcup_k \mathfrak{j}_k. The obvious map j k(𝔧/𝔧 k)j \to \prod_k (\mathfrak{j} / \mathfrak{j}_k) factors through k(𝔧/𝔧 k)\bigoplus_k (\mathfrak{j} / \mathfrak{j}_k), since any element lies in all but finitely many 𝔧 k\mathfrak{j}_k. Now take any injective I kI_k with 0𝔧/𝔧 kI k0 \to \mathfrak{j} / \mathfrak{j}_k \to I_k. The map 𝔧 kI k\mathfrak{j} \to \bigoplus_k I_k cannot extend to the whole RR, since otherwise its image would be contained in a sum of finitely many I kI_k. Therefore, kI k\bigoplus_k I_k is not injective.

232 \Rightarrow 3: Ext R k(A, αX α)\operatorname{Ext}^k_R(A, \bigoplus_\alpha X_\alpha) can be computed by taking an injective resolution of αX α\bigoplus_\alpha X_\alpha. Since direct sums of injective modules are assumed to be injective, we can take a direct sum of injective resolutions of each X αX_\alpha. It remains to note that Hom out of a finitely generated module commutes with arbitrary direct sums.

323 \Rightarrow 2: Follows from the fact that II is injective iff Ext R 1(R/𝔦,I)=0\operatorname{Ext}^1_R(R / \mathfrak{i}, I) = 0 for any ideal 𝔦\mathfrak{i}.

Noetherian and Artinian rings

A dual condition is artinian: an artinian ring is a ring satisfying the descending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian; artinian rings are intuitively much smaller than generic noetherian rings.

References

Last revised on May 25, 2022 at 22:13:01. See the history of this page for a list of all contributions to it.