Chain Conditions: A Reference

Q: What does the DCC on principal left ideals buy that the full DCC does not?

Generality. The full DCC forces rad R to be nilpotent and the ring to be semiprimary; the principal version only forces right T-nilpotence, which admits genuinely infinite-dimensional examples such as (23.22). Since the applications — projective covers, flat implies projective — need only the weaker hypothesis, the weaker hypothesis is the right one.

Executive Summary

Chain conditions are the finiteness hypotheses of ring theory. They come in four independent flavours — ascending or descending, on left ideals or right ideals — and the standard results are careful about which is assumed. The single most quoted implication, artinian implies noetherian, is true for rings on a fixed side and false for modules.

Everything on this page reduces to two facts. Chain conditions pass through short exact sequences $(1.20)$ , which is the workhorse; and a nilpotent radical with semisimple quotient makes ACC, DCC and finite length coincide $(4.15)$ , which is Hopkins–Levitzki.

4Independent conditions

$(1.20)$ The workhorse

$(4.15)$ Hopkins–Levitzki

1939DCC

\Rightarrow

ACC proved

Overview

A ring is left noetherian if it has ACC on left ideals, left artinian if it has DCC on left ideals, and similarly on the right. The four resulting conditions are logically independent except for one implication: left artinian implies left noetherian. That implication is not formal — it was unknown to Noether and Artin and was proved by Hopkins and Levitzki in 1939 — and it fails for modules, where artinian and noetherian are genuinely incomparable.

Left artinianSemiprimaryLeft noetherianEvery left ideal finitely generated

Below the classical conditions sit weaker ones that still support real theorems. The most important is the DCC on principal left ideals, which by Bass's Theorem P $(23.20)$ characterises right perfect rings — see Bass's Theorem P and Perfect and Semiprimary Rings.

Learning Objectives

State ACC and DCC for modules and give the finite-generation and maximal-element reformulations.
Prove that noetherianness and artinianness pass through short exact sequences.
State Hopkins–Levitzki and identify precisely which hypothesis makes it work.
Give a ring that is left artinian and not right artinian, and one that is left noetherian and not left artinian.
Give a module that is artinian and not noetherian.
Choose the correct chain condition for a given theorem, including the principal-ideal variants.

Definitions

DefinitionChain conditions

Let $M$ be a left or right $R$ -module. $M$ is noetherian if the family of submodules of $M$ satisfies the ascending chain condition, and artinian if it satisfies the descending chain condition. A ring $R$ is left noetherian (respectively left artinian) if $_{R} R$ is noetherian (respectively artinian), and right noetherian or right artinian if $R_{R}$ is. Noetherian without qualification means both sides.

ACC: Every ascending chain $M_{1} \subseteq M_{2} \subseteq \dots$ of submodules stabilises; equivalently every nonempty family of submodules has a maximal member.
DCC: Every descending chain $M_{1} \supseteq M_{2} \supseteq \dots$ stabilises; equivalently every nonempty family of submodules has a minimal member.
Finite length: $M$ has a composition series; by $(1.19)$ this holds exactly when $M$ is both noetherian and artinian.
DCC on principal left ideals: Every chain $R a_{1} \supseteq R a_{2} a_{1} \supseteq R a_{3} a_{2} a_{1} \supseteq \dots$ stabilises. Strictly weaker than DCC on left ideals.
Right T-nilpotent: For every sequence $a_{1}, a_{2}, \dots$ in the set, some product $a_{1} a_{2} \dots a_{n}$ vanishes. The chain-condition-free substitute for nilpotence.

Two standard equivalents used constantly: (1.18) M is noetherian iff every submodule is finitely generated; and R is left noetherian iff every left ideal is finitely generated, iff every nonempty family of left ideals has a maximal member.

Core Concepts

Why the two conditions are not symmetric in strength

ACC controls how large submodules can get; DCC controls how small they can get. For modules these are independent: the abelian group $ℤ$ is noetherian and not artinian, and the Prüfer group $ℤ (p^{\infty})$ is artinian and not noetherian. For rings the asymmetry is broken by the presence of an identity, which forces the descending chain condition to interact with the radical.

The four ring conditions and how they interact

Does the row condition imply the column condition, for rings?
	Left noetherian	Right noetherian	Left artinian	Right artinian
Left noetherian	yes	no	no	no
Right noetherian	no	yes	no	no
Left artinian	yes	no	yes	no
Right artinian	no	yes	no	yes

Does the row condition imply the column condition, for rings?

The only non-diagonal “yes” entries are the two Hopkins–Levitzki implications, each staying on its own side. Every other entry is refuted by an explicit ring listed under Comparison.

Weaker conditions that still work

DCC on principal left ideals. Equivalent to right perfectness $(23.20)$ , and implied by DCC on right ideals $(23.21)$ — note the side-switch.
**Right T-nilpotence of $rad R$ .** The exact substitute for nilpotence that makes Nakayama's Lemma work for arbitrary, not merely finitely generated, modules $(23.16)$ .
ACC on right annihilators. Enough for Utumi's argument $(10.29)$ and hence for Levitzki's Theorem and Köthe's conjecture in that class.
Finitely generated over a noetherian ring. By $(1.21)$ a finitely generated module over a left noetherian ring is noetherian, and likewise for artinian.

Key Results

Proposition(1.20)Chain conditions in short exact sequences

Let $N$ be a submodule of a module $M$ . Then $M$ is noetherian if and only if both $N$ and $M / N$ are noetherian; and $M$ is artinian if and only if both $N$ and $M / N$ are artinian. In particular a finite direct sum of noetherian (respectively artinian) modules is noetherian (respectively artinian).

Proof

We give the noetherian case; the artinian case is the same argument with all inclusions reversed.

Necessity. Submodules of $N$ are submodules of $M$ , so ACC is inherited. Submodules of $M / N$ correspond bijectively and order-preservingly to submodules of $M$ containing $N$ , so ACC is inherited there too.

Sufficiency. Let $M_{1} \subseteq M_{2} \subseteq \dots$ be an ascending chain in $M$ . The chains $M_{i} \cap N$ in $N$ and $(M_{i} + N) / N$ in $M / N$ both stabilise, say from index $n$ onwards. Fix $m \in M_{n + 1}$ . Its class in $M / N$ lies in $(M_{n + 1} + N) / N = (M_{n} + N) / N$ , so $m = m' + u$ with $m' \in M_{n}$ and $u \in N$ . Then $u = m - m' \in M_{n + 1} \cap N = M_{n} \cap N \subseteq M_{n}$ , hence $m = m' + u \in M_{n}$ . Therefore $M_{n + 1} = M_{n}$ and the chain stabilises.

Theorem(4.15)Hopkins–Levitzki

Let $R$ be a semiprimary ring: $rad R$ is nilpotent and $R / rad R$ is semisimple. Then for any left $R$ -module $M$ the following are equivalent: (1) $M$ is noetherian; (2) $M$ is artinian; (3) $M$ has a composition series.

Corollary(4.15)Left artinian implies left noetherian

Every left artinian ring is left noetherian. The corresponding statement on the right also holds; there is no implication across sides.

Proof

Let $R$ be left artinian and put $J = rad R$ . By $(4.12)$ , $J$ is nilpotent, say $J^{n} = 0$ , and $R / J$ is left artinian with zero radical, hence semisimple by $(4.14)$ . So $R$ is semiprimary. Consider the filtration

R \supseteq J \supseteq J^{2} \supseteq \dots \supseteq J^{n} = 0 .

Each factor $J^{i} / J^{i + 1}$ is annihilated by $J$ , hence is a module over the semisimple ring $R / J$ , and is therefore a semisimple $R$ -module. It is also artinian, being a subquotient of the artinian module $_{R} R$ . A semisimple artinian module is a finite direct sum of simple modules, so each factor has finite length and in particular is noetherian. Applying $(1.20)$ repeatedly up the filtration, $_{R} R$ is noetherian, that is, $R$ is left noetherian.

Theorem(1.22)Chain conditions for a triangular ring

Let $R$ and $S$ be rings, $M$ an $(R, S)$ -bimodule, and $A = (\begin{matrix} R & M \\ 0 & S \end{matrix})$ . Then $A$ is left noetherian if and only if $R$ and $S$ are left noetherian and $M$ is noetherian as a left $R$ -module. $A$ is right noetherian if and only if $R$ and $S$ are right noetherian and $M$ is noetherian as a right $S$ -module. The same statements hold with noetherian replaced throughout by artinian.

Theorem(23.20), (23.21)Principal chain conditions

A ring $R$ is right perfect — that is, $R / rad R$ is semisimple and $rad R$ is right T-nilpotent — if and only if $R$ satisfies the DCC on principal left ideals. Consequently, if $R$ satisfies the DCC on right ideals then it satisfies the DCC on principal left ideals.

Worked Example

A ring artinian on the left only

Take the field extension $ℚ \subseteq ℝ$ , which has infinite degree, and form the triangular ring

A = (\begin{matrix} ℝ & ℝ \\ 0 & ℚ \end{matrix}) = {(\begin{matrix} a & b \\ 0 & c \end{matrix}) : a, b \in ℝ, c \in ℚ},

(E.1)

The bimodule is $M = ℝ$ , an $(ℝ, ℚ)$ -bimodule.

Apply $(1.22)$ on the left. Both $ℝ$ and $ℚ$ are fields, hence left noetherian and left artinian. As a left $ℝ$ -module, $M = ℝ$ is $ℝ$ itself, which is simple, so it is both noetherian and artinian. Hence $A$ is left noetherian and left artinian.

Apply $(1.22)$ on the right. As a right $ℚ$ -module, $M = ℝ$ is a $ℚ$ -vector space of infinite dimension, so it satisfies neither ACC nor DCC on subspaces. Hence $A$ is neither right noetherian nor right artinian.

Reading it off directly

The left-hand conclusion can be seen without $(1.22)$ . As a left module over itself $A$ has the composition series

A ⊋ (\begin{matrix} ℝ & ℝ \\ 0 & 0 \end{matrix}) ⊋ (\begin{matrix} 0 & ℝ \\ 0 & 0 \end{matrix}) ⊋ 0,

(E.2)

Length $3$ ; by $(1.19)$ finite length is equivalent to being both noetherian and artinian.

On the right, choose an infinite family of nonzero $ℚ$ -subspaces $M_{1}, M_{2}, \dots$ of $ℝ$ whose sum is direct — possible because $\dim_{ℚ} ℝ$ is infinite. Each $(\begin{matrix} 0 & M_{i} \\ 0 & 0 \end{matrix})$ is a right ideal, and their sum is direct, so $A$ contains an infinite direct sum of nonzero right ideals and can satisfy no chain condition on the right.

A second example: noetherian on one side, artinian on neither

Replace the pair $(ℝ, ℚ)$ by $(ℚ, ℤ)$ : set $B = (\begin{matrix} ℚ & ℚ \\ 0 & ℤ \end{matrix})$ . By $(1.22)$ , $B$ is left noetherian, since $ℚ$ and $ℤ$ are noetherian and $ℚ$ is noetherian as a left $ℚ$ -module. It is not right noetherian, because $ℚ$ is not a finitely generated $ℤ$ -module, and it is artinian on neither side, because $ℤ$ is not artinian. This is $(1.23)$ .

Process and Workflow

Which chain condition does your theorem actually need?

Composition series / Jordan–HölderBoth ACC and DCC on the module in question. Over a semiprimary ring either one alone suffices, by

(4.15)

Nilpotence of $rad R$ DCC on left ideals

(4.12)

. ACC alone is not enough:

rad (k [[x]]) = (x)

is not nil and

k [[x]]

is noetherian.

Nil implies nilpotentACC on right ideals suffices, by Levitzki's Theorem

(10.30)

; in fact ACC on right annihilators is enough

(10.29)

Projective covers for all modulesDCC on principal left ideals — that is, right perfectness

(23.20)

. Full DCC is far more than required.

Finitely generated implies finite lengthLeft artinian, via

(1.21)

and Hopkins–Levitzki.

Krull–Schmidt uniquenessNo chain condition on the ring: local endomorphism rings of the summands suffice

(19.21)

Fix the sideDecide whether your modules are left or right modules, and state the chain condition on that side. Half of the errors in this area are side errors.

Try the exact sequence lemma

(1.20)

reduces almost every verification to the case of a filtration with manageable factors.

Look for a triangular presentationIf the ring is triangular,

(1.22)

settles all four conditions by inspecting

R

S

and the bimodule

M

on the appropriate side.

Escalate or weakenIf the full DCC fails, ask whether the principal-ideal version holds; that is often exactly what the theorem needed.

Comparison and Classification

Realising every combination
Ring	Left noeth.	Right noeth.	Left artin.	Right artin.
$M_{n} (D)$ , $D$ a division ring	yes	yes	yes	yes
$ℤ$ , or $k [x]$ with $k$ a field	yes	yes	no	no
$k [[x]]$	yes	yes	no	no
Weyl algebra $A_{1} (k)$ , $char k = 0$	yes	yes	no	no
$(\begin{matrix} ℝ & ℝ \\ 0 & ℚ \end{matrix})$ — $(1.24)$	yes	no	yes	no
$(\begin{matrix} ℚ & ℚ \\ 0 & ℤ \end{matrix})$ — $(1.23)$	yes	no	no	no
$k [x; σ]$ , $σ$ a non-surjective endomorphism of a division ring $k$ — $(1.25)$	yes	no	no	no
$ℤ ⟨ x, y ⟩ / (y^{2}, y x)$ — Dieudonné, $(1.26)$	yes	no	no	no
$k ⟨ x, y ⟩$ , free algebra	no	no	no	no

Modules: the two conditions are independent
Module	Noetherian	Artinian	Comment
$ℤ$ over $ℤ$	yes	no	$(2) ⊋ (4) ⊋ (8) ⊋ \dots$
$ℤ (p^{\infty})$ over $ℤ$	no	yes	Submodules form an ascending chain of finite cyclic groups
$ℤ / n ℤ$ over $ℤ$	yes	yes	Finite, hence finite length
$ℚ$ over $ℤ$	no	no	Neither condition; not finitely generated and no minimal submodule

The second table is the reason the slogan “artinian implies noetherian” must always be attached to rings, not modules.

Relationship Map

DCC on left ideals — Left artinian
- immediately gives
  - $rad R$ nilpotent $(4.12)$
  - $R / rad R$ semisimple $(4.14)$
  - $R$ semiprimary, hence left and right perfect $(23.19)$
  - DCC on principal left ideals
- gives via Hopkins–Levitzki
  - ACC on left ideals
  - $_{R} R$ of finite length
  - Every finitely generated left module of finite length
- does not give
  - any condition on right ideals
  - commutativity of the Wedderburn data

DCC on right idealsDCC on principal left idealsRight perfectSemiperfectSemilocal

The first arrow is $(23.21)$ and the side-switch is genuine; the second is the hard half of Bass's Theorem P. Beyond semilocal there is no chain condition left to weaken, and one changes to the prime and primitive theory instead.

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Computer algebra

Termination guarantees

Gröbner basis algorithms and their noncommutative analogues terminate because of ascending chain conditions on ideals. Where the ACC fails — free algebras, for instance — no general terminating algorithm exists.

Representation theory

Finite length everywhere

Finite-dimensional algebras are artinian, so every finitely generated module has a composition series and Jordan–Hölder multiplicities are well defined. The entire theory of decomposition matrices rests on this.

Control theory

Skew polynomial models

Rings $k [x; σ]$ modelling time-varying linear systems are left noetherian and often not right noetherian; the asymmetry is visible as the difference between forward and backward solvability.

Homological algebra

Dimension theory

Global and finitistic dimension arguments require noetherian hypotheses to guarantee that minimal resolutions exist and are finitely generated in each degree.

Failure Modes and Common Mistakes

Do not assume a subring of a noetherian ring is noetherian: the free algebra $k ⟨ x, y ⟩$ embeds in division rings, which are trivially noetherian.
Do not assume $R$ noetherian makes $R [[x]]$ or an infinite matrix ring noetherian; only finitely many variables and finite matrix size are safe.
Do not read DCC on principal left ideals as a mild variant of DCC: it is satisfied by rings that are very far from artinian, such as the infinite triangular ring of $(23.22)$ .

Best Practices

State the side in every hypothesis and every conclusion, even when the ring at hand happens to be symmetric.
Prefer semiprimary to artinian when nilpotence of the radical is what the proof uses; it is symmetric and strictly weaker.
When verifying a chain condition, look for a finite filtration with tractable factors and invoke $(1.20)$ rather than arguing about chains directly.
Record whether a counterexample is asymmetric by construction — triangular rings and skew polynomial rings are the two standard sources.
For modules, always say which of ACC and DCC is meant; finite length is the safe phrase when both are intended.

Historical Notes and Lessons Learned

1921NoetherThe ascending chain condition is isolated as the right finiteness hypothesis for ideal theory in commutative rings.
1927ArtinThe descending chain condition replaces finite dimension in the structure theory of algebras, extending Wedderburn's classification.
1939Hopkins and LevitzkiIndependently prove that a ring with DCC on left ideals has ACC on left ideals, via nilpotence of the radical. Neither Noether nor Artin had suspected the implication.
1950sOne-sided examplesTriangular rings and skew polynomial rings supply systematic families of rings noetherian or artinian on one side only; Dieudonné's example makes the point with a finitely presented ring.
1960BassThe descending chain condition on principal left ideals is identified as the exact hypothesis for right perfectness, decoupling the useful consequences of DCC from DCC itself.

The pattern to take away: each time a chain condition was weakened, the weaker version turned out to be equivalent to a module-theoretic property that was what the applications had really been using. DCC on left ideals became DCC on principal left ideals became the existence of projective covers.

Quick Reference

NoetherianACC on submodules

\Leftrightarrow

every submodule finitely generated

ArtinianDCC on submodules

Finite lengthNoetherian and artinian

\Leftrightarrow

composition series

(1.19)

Exact sequences

M

noetherian

\Leftrightarrow

N

and

M / N

noetherian

(1.20)

RingsLeft artinian

\Rightarrow

left noetherian; no cross-side implications

RadicalLeft artinian

\Rightarrow

rad R

nilpotent

(4.12)

Hopkins–LevitzkiSemiprimary

\Rightarrow

noetherian

=

artinian

=

finite length

(4.15)

BassDCC on principal left ideals

\Leftrightarrow

right perfect

(23.20)

Triangular

(1.22)

decides all four conditions from

R

S

and

M

Standard witnesses
Needed	Ring or module
Noetherian, not artinian	$ℤ$
Artinian module, not noetherian	$ℤ (p^{\infty})$ over $ℤ$
Left artinian, not right artinian	$(\begin{matrix} ℝ & ℝ \\ 0 & ℚ \end{matrix})$
Left noetherian, not right noetherian	$k [x; σ]$ , $σ$ non-surjective
Neither, on either side	$k ⟨ x, y ⟩$
DCC on principal left ideals, not artinian	Lam's infinite triangular ring $(23.22)$

Frequently Asked Questions

Why is “artinian implies noetherian” true for rings but false for modules?

The proof uses the identity element and the radical: DCC on left ideals makes $rad R$ nilpotent, the powers of the radical give a finite filtration of $R$ , and each factor is a semisimple artinian module hence of finite length. A general module has no such intrinsic filtration, and the Prüfer group $ℤ (p^{\infty})$ shows that none can be manufactured.

Is left noetherian equivalent to right noetherian for any natural class of rings?

Yes for commutative rings trivially, and for rings that are finitely generated modules over a commutative noetherian centre. It is not equivalent in general: $k [x; σ]$ with $σ$ a non-surjective endomorphism of a division ring is left noetherian and not right noetherian $(1.25)$ , and Dieudonné's ring $ℤ ⟨ x, y ⟩ / (y^{2}, y x)$ makes the point with a finitely presented example $(1.26)$ .

What does the DCC on principal left ideals buy that the full DCC does not?

Generality. The full DCC forces $rad R$ to be nilpotent and the ring to be semiprimary; the principal version only forces right T-nilpotence, which admits genuinely infinite-dimensional examples such as $(23.22)$ . Since the applications — projective covers, flat implies projective — need only the weaker hypothesis, the weaker hypothesis is the right one.

Does the semiprimary hypothesis in Hopkins–Levitzki matter, or is artinian enough?

It matters, because semiprimary is strictly weaker and symmetric. A trivial extension $k \oplus V$ with $\dim_{k} V$ infinite and $V^{2} = 0$ is semiprimary and neither noetherian nor artinian, yet $(4.15)$ still applies to its modules and equates the two chain conditions on any one of them.

How do chain conditions behave under standard constructions?

Matrix rings preserve all four conditions on the corresponding side. Finite direct products preserve them. Polynomial rings in finitely many commuting variables preserve noetherianness by the Hilbert Basis Theorem but destroy artinianness. Subrings and infinite matrix rings preserve nothing, and quotients preserve everything.

Where do chain conditions get used in the classification theorems?

In three places, and it is worth learning to spot them. They make the radical nilpotent, they collapse the Density Theorem's dense subring to the full endomorphism ring, and they supply composition series so that Jordan–Hölder multiplicities exist. If a proof appears to use finiteness, it is almost certainly one of these three.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §1 (pp. 18–24), §4 (pp. 58–61) and §23 (pp. 352–357).
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964, Chapter IV.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §10–§11 and §28.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics 30, American Mathematical Society, revised edition, 2001.
H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the American Mathematical Society 95 (1960), 466–488.

AI Suggested Questions

Construct a ring that is right artinian but not left noetherian, or prove that none exists.
How do the four chain conditions behave under Morita equivalence?
What is the weakest chain condition under which Krull–Schmidt uniqueness still holds?
Give a complete proof that $k [x; σ]$ is left noetherian and not right noetherian when $σ$ is a non-surjective endomorphism of a division ring.
Which chain conditions are inherited by corner rings $e R e$ and by fixed rings under a finite group action?
Explain the relationship between the DCC on principal left ideals and the existence of projective covers.
Is there an ascending analogue of Bass's Theorem P characterising some natural class of rings?

Chain Conditions Reference

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Why the two conditions are not symmetric in strength

The four ring conditions and how they interact

Weaker conditions that still work

Key Results

Worked Example

A ring artinian on the left only

Reading it off directly

A second example: noetherian on one side, artinian on neither

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Termination guarantees

Finite length everywhere

Skew polynomial models

Dimension theory

Failure Modes and Common Mistakes

Best Practices

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Why the two conditions are not symmetric in strength

The four ring conditions and how they interact

Weaker conditions that still work

06Key Results

07Worked Example

A ring artinian on the left only

Reading it off directly

A second example: noetherian on one side, artinian on neither

08Process and Workflow

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

Termination guarantees

Finite length everywhere

Skew polynomial models

Dimension theory

12Failure Modes and Common Mistakes

13Best Practices

14Historical Notes and Lessons Learned

15Quick Reference

16Frequently Asked Questions

17Related KEVOS Topics

18References

19AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Worked Example

Process and Workflow

Comparison and Classification

Relationship Map

Applications and Industry Use

Failure Modes and Common Mistakes

Best Practices

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions