Noetherian Modules and the Ascending Chain Condition

Q: Is every finitely generated module Noetherian?

No. Over a general ring the two are different, and K[x_1,x_2, … ] as a module over itself is the standard counterexample: it is generated by 1 and contains an ideal that no finite set generates. Over a Noetherian ring the two conditions do coincide, which is why the distinction so often goes unnoticed.

Q: Why prefer the chain condition to the definition?

Because it converts a statement about all submodules into a statement about processes. Any construction that repeatedly enlarges something can be shown to halt by exhibiting the enlargement as a strictly ascending chain. That is how termination of Gröbner basis computation is proved, and no direct argument from finite generation is as convenient.

Q: Does the maximal condition require the axiom of choice?

Deriving it from the ascending chain condition uses dependent choice, since the strictly ascending chain is built by infinitely many successive selections. The other two implications in the equivalence are choice-free. In ordinary mathematical practice the distinction is ignored, but it is real.

Q: Are Noetherian and Artinian related?

They are logically independent for modules: Z is Noetherian and not Artinian, the Prüfer group Z(p^ ∈ fty) is Artinian and not Noetherian. A module has finite length exactly when it is both. For rings there is an asymmetry: by the Hopkins-Levitzki theorem a left Artinian ring is left Noetherian, but not conversely.

Q: If every submodule needs finitely many generators, is there a bound on how many?

Not in general. In K[x_1,x_2] the ideal generated by all monomials of degree k needs k+1 generators, and k is unbounded. The Weyl algebra is unusual in admitting such a bound - every left ideal of A_n is generated by two elements - but that is a deep theorem of Stafford, not a consequence of being Noetherian.

Q: Does the property depend on which side the ring acts?

For modules the two theories are mirror images and everything on this page holds verbatim for right modules. For rings the sides can differ: there exist rings that are left Noetherian and not right Noetherian. The Weyl algebra is Noetherian on both sides.

Q: Why does D-module theory need this rather than finite dimension over K?

Because no non-zero A_n -module is finite dimensional over K - the operators x and ∂ cannot both act on a non-zero finite-dimensional space, since their commutator would have trace 0 and trace simultaneously. The Noetherian condition is the usable finiteness hypothesis in its place; see the page on this obstruction .

Q: How is the Weyl algebra shown to be Noetherian?

Not directly. One filters A_n by the Bernstein filtration, observes that the associated graded ring is a commutative polynomial ring in 2n variables and hence Noetherian by Hilbert's basis theorem, and then transfers the property back down to A_n . That transfer is the key theorem of this chapter .

Overview

Almost every module that appears in D-module theory is finitely generated, and finite generation alone is a weak hypothesis. It is not inherited by submodules: a module can be generated by a single element and still contain a submodule that no finite list generates. The standard example is the polynomial ring $R = K[x_1, x_2, x_3, … ]$ in countably many variables. As a module over itself it is cyclic, generated by $1$ ; the ideal $I = ⟨ x_1, x_2, … ⟩$ is a submodule of it that is not finitely generated, because any finite list of polynomials mentions only finitely many of the variables.

The repair is to build the missing stability into the hypothesis. A module is Noetherian when every submodule is finitely generated. This single condition is what makes the rest of the theory work: it is what allows induction on submodules, what makes algorithms terminate, and what will eventually let us attach a well-behaved good filtration, a dimension and a multiplicity to a module over the Weyl algebra.

The page proves the standard equivalence: being Noetherian is the same as the ascending chain condition (increasing chains of submodules stop) and the same as the maximal condition (every non-empty family of submodules has a maximal member). The three formulations are used interchangeably in practice, and each is convenient for a different kind of argument: finite generation for constructing things, the chain condition for proving termination, the maximal condition for extremal arguments of the form "take a counterexample maximal with respect to ...".

This is the foundation stone for the rest of the chapter. From here, the behaviour of the property in short exact sequences gives the closure rules, Noetherian rings supply the examples through Hilbert's basis theorem, and the graded-to-filtered transfer carries the property from a polynomial ring up to the Weyl algebra.

Definition

Throughout, $R$ is an associative ring with $1$ and modules are left $R$ -modules; every statement has a right-handed twin obtained by reversing the side on which $R$ acts.

Finitely generated module

A left $R$ -module $M$ is finitely generated if there are $u_1, … , u_t ∈ M$ with

M = Ru_1 + … + Ru_t = a_1 u_1 + … + a_t u_t : a_i ∈ R .

Equivalently, there is a surjective homomorphism $R^t ↠ M$ of left $R$ -modules.

Noetherian moduleCoutinho (8.1)

A left $R$ -module $M$ is Noetherian if every submodule $N ⊂ eq M$ is finitely generated. Taking $N = M$ shows that a Noetherian module is itself finitely generated.

The two chain formulations

$M$ satisfies the ascending chain condition (ACC) if for every chain of submodules

N_1 ⊂ eq N_2 ⊂ eq N_3 ⊂ eq … ⊂ eq M

there is a $k$ with $N_i = N_k$ for all $i ≥ k$ . $M$ satisfies the maximal condition if every non-empty set $S$ of submodules of $M$ has an element $L ∈ S$ that is properly contained in no element of $S$ .

The family must be non-empty

The maximal condition is stated in some texts, including the source for this collection, without the words "non-empty". As written that is false for trivial reasons: the empty family of submodules has no member at all, maximal or otherwise. The hypothesis $S ≠ ∅$ is intended and is used in every application. It is restored here.

Core Concepts

Why finite generation alone is fragile

Finite generation passes to quotients without trouble: if $M = Ru_1 + … + Ru_t$ and $M ↠ M/N$ , then the images of the $u_i$ generate $M/N$ . It does not pass to submodules. The failure is not exotic; it is exactly what happens when the ring itself is too large, and it is the reason the definition quantifies over all submodules rather than assuming finite generation of $M$ alone.

Three views of the same condition

The three formulations say the same thing in three different languages.

Finite generation of submodules is the constructive form. It is what you use when you need to produce data: a finite list of generators for a submodule, a presentation, a Gröbner basis.
The ascending chain condition is the dynamic form. It says that a process which repeatedly enlarges a submodule cannot run forever. This is the form that proves algorithms terminate.
The maximal condition is the extremal form. It licenses arguments that begin "among all submodules with property $P$ , choose one that is maximal", which is how many structure theorems are set up.

Noetherian induction

The maximal condition is what makes Noetherian induction valid. To prove that every submodule of a Noetherian module $M$ has a property $P$ , it suffices to show that if every submodule strictly containing $N$ has $P$ , then $N$ has $P$ . If the set of submodules failing $P$ were non-empty it would have a maximal member, and that member would then have $P$ after all. This is the reverse of ordinary induction: one climbs the lattice of submodules rather than the integers.

Where the property comes from in practice

Almost no module is checked against the definition directly. The property is imported: one shows the ring is Noetherian, and then every finitely generated module over it is automatically Noetherian. That chain of reasoning is completed on the Noetherian rings page, and for the Weyl algebra $A_n$ it is completed on the page proving P1 Noetherian.

Construction and Proof

The three conditions are equivalentCoutinho (8.1.1)

Let $M$ be a left $R$ -module. The following are equivalent.

Every submodule of $M$ is finitely generated.
For every ascending chain $N_1 ⊂ eq N_2 ⊂ eq …$ of submodules of $M$ there is $k$ with $N_i = N_k$ for all $i ≥ k$ .
Every non-empty set $S$ of submodules of $M$ has a maximal element.

Proof

(1) implies (2)

Let $N_1 ⊂ eq N_2 ⊂ eq …$ be a chain and set $Q = ⋃ _i ≥ 1 N_i$ . A union of a chain of submodules is a submodule: any two elements of $Q$ lie in a common $N_i$ because the family is totally ordered, so $Q$ is closed under addition, and it is visibly closed under the action of $R$ . By (1), $Q = Ru_1 + … + Ru_t$ for finitely many $u_1, … , u_t$ . Each $u_j$ lies in some $N_i_j$ ; take $k = _j i_j$ , so all the $u_j$ lie in $N_k$ . Then for $i ≥ k$ ,

Q = Ru_1 + … + Ru_t ⊂ eq N_k ⊂ eq N_i ⊂ eq Q,

forcing equality throughout. The chain is constant from step $k$ on.

(2) implies (3)

Suppose $S ≠ ∅$ has no maximal element. Pick $N_1 ∈ S$ . Since $N_1$ is not maximal there is $N_2 ∈ S$ with $N_1 ⊂ neq N_2$ ; since $N_2$ is not maximal there is $N_3 ∈ S$ with $N_2 ⊂ neq N_3$ ; and so on. This produces a strictly ascending chain that never becomes constant, contradicting (2).

(3) implies (1)

Let $N ⊂ eq M$ be a submodule and let $S$ be the set of all finitely generated submodules of $N$ . It is non-empty, since $0 ∈ S$ , so by (3) it has a maximal element $L$ . If $L ≠ N$ , choose $u ∈ N ∖ L$ . Then $L + Ru$ is finitely generated, is contained in $N$ , and properly contains $L$ , contradicting maximality. Hence $L = N$ and $N$ is finitely generated. $□$

A set-theoretic footnote

The step from (2) to (3) makes infinitely many successive choices and therefore uses the axiom of dependent choice; the same is true of most textbook proofs of this implication. The implications (1) $\Rightarrow$ (2) and (3) $\Rightarrow$ (1) are choice-free. Nothing in this collection turns on the distinction, but it is worth knowing that the equivalence is not a purely finitary statement.

Key Equations

The submodule generated by a subset $X ⊂ eq M$ is the smallest submodule containing it:

⟨ X ⟩ = a_1 u_1 + … + a_r u_r : r ≥ 0, a_i ∈ R, u_i ∈ X .

(8.1)

The module is Noetherian precisely when the assignment $N \mapsto$ (a finite generating set) can always be made, that is when

for all N ⊂ eq M there exist u_1, … ,u_t ∈ N with N = ⟨ u_1, … ,u_t ⟩ .

(8.2)

The chain condition is the statement that the following union is always attained at a finite stage:

⋃ _i ≥ 1 N_i = N_k for some k, N_1 ⊂ eq N_2 ⊂ eq …

(8.3)

For the Weyl algebra the relevant instance, proved later in the chapter, reads

M finitely generated over A_n ; ⟹ ; M Noetherian,

(8.4)

valid because $A_n$ is a Noetherian ring; the implication is false over a general ring.

Variable Definitions

$R$: an associative ring with identity; modules are left $R$ -modules unless stated otherwise
$M$: the module under discussion
$N$ , $L$ , $P$: submodules of $M$
$N_i$: the $i$ -th term of an ascending chain of submodules
$Q$: the union $⋃ _i N_i$ of an ascending chain, itself a submodule
$S$: a non-empty set of submodules of $M$ , in the maximal condition
$⟨ X ⟩$: the submodule of $M$ generated by a subset $X$
$K$: a field, usually of characteristic zero
$A_n$: the $n$ -th Weyl algebra over $K$ , the running example of a non-commutative Noetherian ring

Properties and Behaviour

Immediate consequences of the definition

A Noetherian module is finitely generated (take $N = M$ ).
Every submodule of a Noetherian module is Noetherian, since its submodules are among those of $M$ .
Every quotient of a Noetherian module is Noetherian: submodules of $M/N$ are of the form $L/N$ with $N ⊂ eq L ⊂ eq M$ , and generators of $L$ map onto generators of $L/N$ .
A simple module, and more generally a module of finite length, is Noetherian.

Finite sums

If $M = N + N'$ with $N$ and $N^{'}$ Noetherian submodules, then $M$ is Noetherian; by induction, a finite sum of Noetherian submodules is Noetherian, and a finite direct sum of Noetherian modules is Noetherian. The proof runs through the short exact sequence $0 → N → M → M/N → 0$ and is given on the exact sequence page.

Infinite direct sums fail

Finiteness in the previous statement cannot be dropped. Take $M = ⨁ _i ≥ 1 K$ as a $K$ -module: each summand is Noetherian, but the partial sums form a strictly ascending chain with no last term. Equivalently, $K[x_1,x_2, … ]$ above is a countable union of Noetherian pieces $K[x_1, … ,x_N]$ and is not Noetherian.

Noetherian induction

Let $M$ be Noetherian and let $P$ be a property of submodules. If for every submodule $N$ the implication " $P$ holds for every submodule strictly containing $N$ " $\Rightarrow$ " $P$ holds for $N$ " is valid, then $P$ holds for every submodule of $M$ . Indeed the set of submodules failing $P$ , if non-empty, has a maximal element $N_0$ ; every submodule strictly containing $N_0$ then satisfies $P$ , so $N_0$ does too - a contradiction.

Examples and Special Cases

Finite-dimensional vector spaces

Let $R = K$ be a field and let $V$ be a $K$ -vector space. Every subspace of $V$ is spanned by a basis, so $V$ is Noetherian if and only if every subspace has a finite basis, that is if and only if $_K V < ∈ fty$ . A chain of subspaces of a space of dimension $d$ has at most $d+1$ distinct terms, which is the chain condition with an explicit bound.

An infinite-dimensional vector space is not Noetherian

In $V = K[x]$ viewed as a $K$ -vector space, the subspaces $V_i$ of polynomials of degree less than $i$ form a strictly ascending chain that never stops, and $V$ itself is not finitely generated over $K$ . The claim, made loosely in some texts, that "vector spaces are Noetherian modules" is true only in the finite-dimensional case. Note the contrast: the same set $K[x]$ is Noetherian as a module over $K[x]$ , and is Noetherian as a module over the Weyl algebra $A_1$ because it is simple there. Being Noetherian is a property of the pair (ring, module), never of the underlying set.

Principal ideal domains

$ℤ$ as a $ℤ$ -module and $K[x]$ as a $K[x]$ -module are Noetherian: every submodule is an ideal, and every ideal is generated by one element. So is every ideal of $K[x]$ regarded as a $K[x]$ -module, since its submodules are again ideals.

The rationals over the integers

$ℚ$ is not a Noetherian $ℤ$ -module. The submodules generated by $2^-i$ form a strictly ascending chain

Z ⋅ , 12 ⊂ neq Z ⋅ , 14 ⊂ neq Z ⋅ , 18 ⊂ neq …

which never becomes constant, since $2^-i-1 ∉ Z ⋅ 2^-i$ . Here the ring $ℤ$ is Noetherian while the module is not: what fails is finite generation of $ℚ$ , not any property of $ℤ$ .

Simple modules

A simple module has only the submodules $0$ and $M$ , both finitely generated ( $M$ by any non-zero element), so it is Noetherian. This is how P3 as an P4-module is seen to be Noetherian without any machinery: it is simple.

Modules over the Weyl algebra

Every finitely generated $A_n$ -module is Noetherian. This is not obvious from the definition; it follows from the fact that $A_n$ is a Noetherian ring, which is the main theorem of this chapter.

Worked Example

A cyclic module that is not Noetherian, checked against all three conditions

Step 1 - the module
Let $R = K[x_1, x_2, x_3, … ]$ be the polynomial ring in countably many variables over a field $K$ : its elements are polynomials, so each one involves only finitely many of the variables. Regard $R$ as a left module over itself. It is cyclic, generated by the single element $1$ , since $R ⋅ 1 = R$ .

Let $I = ⟨ x_1, x_2, x_3, … ⟩$ be the ideal generated by all the variables, that is the set of polynomials with zero constant term. It is a submodule of $R$ .
Step 2 - condition (1) fails: $I$ is not finitely generated
Suppose $I = ⟨ f_1, … , f_r ⟩$ . Each $f_j$ involves finitely many variables, so there is an $N$ with $f_1, … , f_r ∈ K[x_1, … ,x_N]$ . Define a $K$ -algebra homomorphism

$ψ : R ⟶ K[t], ψ(x_N+1) = t, ψ(x_i) = 0 for i ≠ N+1 .$

Each $f_j$ lies in $K[x_1, … ,x_N]$ and has zero constant term, so $ψ(f_j) = f_j(0, … ,0) = 0$ . Hence $ψ$ kills every element of $⟨ f_1, … ,f_r ⟩$ . But $ψ(x_N+1) = t ≠ 0$ and $x_N+1 ∈ I$ , so $x_N+1 ∉ ⟨ f_1, … ,f_r ⟩$ and the supposed generating set fails.
Step 3 - condition (2) fails: an infinite strict chain
Put $J_k = ⟨ x_1, … , x_k ⟩$ for $k ≥ 1$ . Clearly $J_k ⊂ eq J_k+1$ . The containment is strict: apply the same map $ψ$ , this time sending $x_k+1 ↦ t$ and all other variables to $0$ . It kills $x_1, … , x_k$ , hence kills $J_k$ , while $ψ(x_k+1) = t ≠ 0$ . So

$J_1 ⊂ neq J_2 ⊂ neq J_3 ⊂ neq …$

is a strictly ascending chain of submodules of $R$ that never becomes constant.
Step 4 - condition (3) fails: a family with no maximal element
Take $S = J_k : k ≥ 1$ , which is non-empty. By Step 3 every $J_k$ is properly contained in $J_k+1 ∈ S$ , so no member of $S$ is maximal. All three conditions of the equivalence theorem fail together, exactly as the theorem predicts.
Step 5 - contrast: finitely many variables
Replace $R$ by $K[x_1,x_2]$ . Now the analogous chain terminates immediately: $⟨ x_1 ⟩ ⊂ neq ⟨ x_1, x_2 ⟩$ and there is nothing further to adjoin. Indeed by Hilbert's basis theorem every ideal of $K[x_1,x_2]$ is finitely generated. Note that no uniform bound on the number of generators exists even here: the ideal $I_k ⊂ eq K[x_1,x_2]$ generated by all monomials of degree $k$ needs exactly $k+1$ generators, namely $x_1^k, x_1^k-1x_2, … , x_2^k$ , and no fewer. Noetherian means "finitely many for each", not "boundedly many across all".

Result

$R = K[x_1, x_2, … ]$ is a cyclic - hence finitely generated - module over itself that is not Noetherian: the ideal of polynomials with zero constant term needs infinitely many generators, the chain $⟨ x_1 ⟩ ⊂ neq ⟨ x_1,x_2 ⟩ ⊂ neq …$ never stops, and the family of those ideals has no maximal member. Finite generation of $M$ therefore does not imply finite generation of the submodules of $M$ .

Applications and Industry Use

In a mathematics topic, this section covers downstream use inside mathematics, computing and engineering rather than a manufactured product.

Termination of Gröbner basis algorithms. Buchberger's algorithm enlarges a set of leading terms at every step; the ascending chain condition in the polynomial ring is exactly what guarantees the process stops. The same argument, applied in the Weyl algebra, underlies every computational D-module package.
Existence of good filtrations. The dimension theory of the next chapters requires that a finitely generated $A_n$ -module carries a good filtration, and that submodules inherit a usable one. The Noetherian property of $A_n$ is what supplies this.
Finite presentations. A module over a Noetherian ring given by generators has a finitely generated module of relations, so it has a finite presentation matrix. Systems of linear partial differential equations can therefore be stored and manipulated as finite data; see presentation matrices.
Structure theorems. Primary decomposition, the existence of associated primes, and the theory of length all rest on the maximal condition, which supplies the extremal objects those theorems are built around.
Symbolic summation and integration. The closure properties that make Zeilberger's algorithm work are proved by finite-dimensionality arguments over a Noetherian base; without the chain condition, the ansatz search would not be known to terminate.

Computational Notes

Read this as the manufacturing section of the template: how the object is actually built by machine, at what cost, and where the computation stops being decidable.

The chain condition is a termination certificate, and it is a purely qualitative one.

An algorithm that grows a submodule - adding a syzygy, adjoining an S-polynomial's remainder, saturating an ideal - terminates because the submodules it produces form a strictly ascending chain until the algorithm stops.
The proof supplies no bound on the number of steps. Bounds must come from elsewhere: degree bounds, Hilbert function bounds, or complexity results specific to the algorithm.
Testing whether a given submodule is finitely generated is not usually a computation performed at run time. The property is inherited from the ring, established once and for all.
In implementations, the practical form of the condition is that leading-term ideals stabilise, which reduces the question to Dickson's lemma on monomial ideals - a finitary combinatorial statement that can be checked directly.

Worst-case behaviour is severe: strictly ascending chains of ideals in $K[x_1, … ,x_n]$ with degree bounds can have length growing faster than any primitive recursive function of the bound. Systems such as Macaulay2, Singular and SageMath rely on the chain condition for correctness of termination and on heuristics for actual performance.

Limits of Validity

Noetherian is not a property of a set. $K[x]$ is Noetherian over $K[x]$ and over $A_1$ , and not Noetherian over $K$ . The ring must always be named.
Noetherian does not bound the number of generators. Each submodule needs finitely many generators, but there need be no bound valid for all submodules at once; the ideals $I_k ⊂ eq K[x_1,x_2]$ generated by the monomials of degree $k$ require $k+1$ generators. (The Weyl algebra is exceptional here: every left ideal of $A_n$ is generated by two elements, a much harder theorem due to Stafford.)
The chain condition gives no effective bound. Knowing a chain must stop says nothing about where. Chains in a polynomial ring can be extremely long, and the worst-case bounds are Ackermannian in the degree. Termination and efficiency are different questions.
Left and right differ in general. A ring can be left Noetherian and not right Noetherian. For $A_n$ both hold, because the transposition anti-automorphism converts one side into the other, but this must be argued, not assumed.
Infinite constructions escape. Arbitrary direct sums, direct limits, infinite tensor constructions and localisations at large multiplicative sets need not preserve the property, even though finite versions of all of them do.

Failure Modes and Common Mistakes

Confusing "finitely generated" with "Noetherian"

This is the mistake the definition exists to prevent. Every Noetherian module is finitely generated; the converse fails, and the worked example above is the standard witness. The two coincide only when the base ring is Noetherian, and then only as a theorem. Whenever a proof passes from a module to a submodule and continues to assume finite generation, check that the ring has been assumed Noetherian.

Reading "maximal" as "largest"

A maximal element $L$ of a family $S$ is one that is properly contained in no member of $S$ . It need not contain the other members, and $S$ may have many maximal elements. In $ℤ$ , the family of proper non-zero ideals has maximal elements $p ℤ$ for every prime $p$ , none of which contains any other. Arguments that quietly upgrade "maximal" to "largest" are wrong.

Believing the union of a chain of finitely generated modules is finitely generated

The union of an ascending chain of submodules is a submodule, and that is all. Each $K[x_1, … ,x_N]$ is Noetherian and their union is not. What the ascending chain condition provides is that the union is attained, that is equal to one of the terms - and that is a hypothesis about $M$ , not a general fact.

Assuming the property survives an infinite limit

In D-module theory several important modules are built as direct limits - for example $K[x][1/f]$ as a limit along multiplication by $f$ , or modules of germs of functions. A direct limit of Noetherian modules is generally not Noetherian, and when such a limit does turn out to be a finitely generated $A_n$ -module that is a substantive theorem about the specific construction, not an instance of a closure property.

Checking only finitely many steps of a chain

A chain that appears to have stabilised over the first several terms may resume growing. Verifying the ascending chain condition means proving it for all chains, which in practice always means invoking a structural theorem such as Hilbert's basis theorem, never inspecting examples.

Historical Notes

David Hilbert proved in 1890 that every ideal of a polynomial ring in finitely many variables over a field is finitely generated - the basis theorem, proved to settle a question in invariant theory. The proof was famously non-constructive and drew Gordan's remark that it was theology rather than mathematics; the constructive content was recovered only decades later, in the work that became the theory of Gröbner bases.

Emmy Noether isolated the abstract condition in her 1921 paper Idealtheorie in Ringbereichen, where the ascending chain condition appears as an axiom in its own right and is used to derive primary decomposition. The insight was that the chain condition, not any special feature of polynomial rings, carried the arguments. The adjective "Noetherian" came into use in the following decades, largely through the Bourbaki school.

For non-commutative rings the theory developed later and more slowly; left and right conditions separate, and rings that are Noetherian on one side only exist. That the Weyl algebra is Noetherian on both sides is proved in this chapter by lifting the property from a commutative polynomial ring through the Bernstein filtration, a technique that goes back to the filtered-graded arguments of the 1960s and is now standard for enveloping algebras and rings of differential operators alike.

Comparison

The Noetherian condition sits in a small family of finiteness conditions. The table separates them; none of the implications marked "no" can be reversed.

Finiteness conditions on a module, compared.
Condition	Says	Implies finitely generated?	Typical example	Typical failure
Finitely generated	$M = Ru_1 + … + Ru_t$	yes, by definition	$K[x_1,x_2, … ]$ over itself	$ℚ$ over $ℤ$
Noetherian	every submodule finitely generated	yes	$ℤ$ over $ℤ$	$K[x_1,x_2, … ]$ over itself
Artinian	descending chains stop	no	$Z(p^ ∈ fty)$ over $ℤ$	$ℤ$ over $ℤ$
Finite length	a finite composition series exists	yes	$K^d$ over $K$	$ℤ$ over $ℤ$
Finite dimension over $K$	$_K M < ∈ fty$	yes when $K ⊂ eq R$	$K^d$ over $K$	any non-zero $A_n$ -module

Two entries deserve comment. The Prüfer group $Z(p^ ∈ fty) = Z[1/p]/Z$ is Artinian but not Noetherian, and $ℤ$ is Noetherian but not Artinian, so the two chain conditions are genuinely independent; a module has finite length exactly when it satisfies both. The last row records the fact that no non-zero module over $A_n$ is finite dimensional over $K$ , so that condition is useless in D-module theory, which is precisely why the Noetherian condition is used instead.

Key Takeaways

Key points

A module is Noetherian when every submodule is finitely generated - not merely when the module itself is.
This is equivalent to the ascending chain condition and to the maximal condition on non-empty families of submodules.
Finite generation alone is not inherited by submodules: $K[x_1,x_2, … ]$ is cyclic over itself and not Noetherian.
Submodules and quotients of Noetherian modules are Noetherian; finite sums are Noetherian; infinite direct sums are not.
The maximal condition powers Noetherian induction and extremal arguments; the chain condition powers termination proofs.
Whether a module is Noetherian depends on the ring acting: $K[x]$ is Noetherian over $K[x]$ and over $A_1$ , but not over $K$ .
In this collection the property is imported from the ring: $A_n$ is Noetherian, so every finitely generated $A_n$ -module is.

FAQs

Is every finitely generated module Noetherian?

No. Over a general ring the two are different, and $K[x_1,x_2, … ]$ as a module over itself is the standard counterexample: it is generated by $1$ and contains an ideal that no finite set generates. Over a Noetherian ring the two conditions do coincide, which is why the distinction so often goes unnoticed.

Why prefer the chain condition to the definition?

Because it converts a statement about all submodules into a statement about processes. Any construction that repeatedly enlarges something can be shown to halt by exhibiting the enlargement as a strictly ascending chain. That is how termination of Gröbner basis computation is proved, and no direct argument from finite generation is as convenient.

Does the maximal condition require the axiom of choice?

Deriving it from the ascending chain condition uses dependent choice, since the strictly ascending chain is built by infinitely many successive selections. The other two implications in the equivalence are choice-free. In ordinary mathematical practice the distinction is ignored, but it is real.

Are Noetherian and Artinian related?

They are logically independent for modules: $ℤ$ is Noetherian and not Artinian, the Prüfer group $Z(p^ ∈ fty)$ is Artinian and not Noetherian. A module has finite length exactly when it is both. For rings there is an asymmetry: by the Hopkins-Levitzki theorem a left Artinian ring is left Noetherian, but not conversely.

If every submodule needs finitely many generators, is there a bound on how many?

Not in general. In $K[x_1,x_2]$ the ideal generated by all monomials of degree $k$ needs $k+1$ generators, and $k$ is unbounded. The Weyl algebra is unusual in admitting such a bound - every left ideal of $A_n$ is generated by two elements - but that is a deep theorem of Stafford, not a consequence of being Noetherian.

Does the property depend on which side the ring acts?

For modules the two theories are mirror images and everything on this page holds verbatim for right modules. For rings the sides can differ: there exist rings that are left Noetherian and not right Noetherian. The Weyl algebra is Noetherian on both sides.

Why does D-module theory need this rather than finite dimension over

K

Because no non-zero $A_n$ -module is finite dimensional over $K$ - the operators $x$ and $∂$ cannot both act on a non-zero finite-dimensional space, since their commutator would have trace $0$ and trace $dim$ simultaneously. The Noetherian condition is the usable finiteness hypothesis in its place; see the page on this obstruction.

How is the Weyl algebra shown to be Noetherian?

Not directly. One filters $A_n$ by the Bernstein filtration, observes that the associated graded ring is a commutative polynomial ring in $2n$ variables and hence Noetherian by Hilbert's basis theorem, and then transfers the property back down to $A_n$ . That transfer is the key theorem of this chapter.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 8 §1, definition and Theorem (8.1.1).
E. Noether, Idealtheorie in Ringbereichen, Mathematische Annalen 83 (1921), 24-66 - the paper in which the ascending chain condition is isolated as an axiom.
D. Hilbert, Über die Theorie der algebraischen Formen, Mathematische Annalen 36 (1890), 473-534 - the basis theorem.
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969 - Ch. 6, chain conditions for modules.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics 30, revised edition, American Mathematical Society, 2001 - Ch. 1, for the non-commutative theory and the left/right distinction.
J.-E. Björk, Rings of Differential Operators, North-Holland Mathematical Library 21, North-Holland, 1979 - Ch. 1, filtered rings and their Noetherian properties.
J. T. Stafford, Module structure of Weyl algebras, Journal of the London Mathematical Society (2) 18 (1978), 429-442 - the two-generator theorem for left ideals of $A_n$ .
ISO 80000-2:2019, Quantities and units - Part 2: Mathematics, International Organization for Standardization.

AI Suggested Questions

Show that a module is Noetherian if and only if every submodule generated by countably many elements is finitely generated.
Give an example of a ring that is left Noetherian but not right Noetherian, and explain what breaks.
Prove that the free algebra $K ⟨ x, y ⟩$ is not left Noetherian by exhibiting a strictly ascending chain of left ideals.
Work out which submodules of $ℚ$ over $ℤ$ are finitely generated, and describe them all.
Show that a module of finite length is both Noetherian and Artinian, and that the converse holds.
Explain precisely where dependent choice is used in the passage from the ascending chain condition to the maximal condition.
Verify that the ideal of $K[x_1,x_2]$ generated by the monomials of degree $k$ cannot be generated by fewer than $k+1$ elements.

Noetherian Modules and the Ascending Chain Condition

Overview

Definition

Core Concepts

Why finite generation alone is fragile

Three views of the same condition

Noetherian induction

Where the property comes from in practice

Construction and Proof

(1) implies (2)

(2) implies (3)

(3) implies (1)

Key Equations

Variable Definitions

Properties and Behaviour

Examples and Special Cases

Worked Example

Applications and Industry Use

Computational Notes

Limits of Validity

Failure Modes and Common Mistakes

Historical Notes

Comparison

Key Takeaways

Key points

FAQs

Related Engineering Topics

References

AI Suggested Questions

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