The Hopkins–Levitzki Theorem

Q: Where exactly is nilpotency of the radical used?

Only to make the filtration finite. If J were merely nil or T-nilpotent, the chain M JM … need not reach zero in finitely many steps and the concatenation of composition series would fail. That is precisely why perfect rings, where J is T-nilpotent, satisfy a weaker theorem.

Executive Summary

Artin's 1927 extension of Wedderburn's theory assumed both chain conditions. Hopkins and Levitzki showed independently in 1939 that one of them is redundant: for rings with an identity, the descending chain condition on left ideals implies the ascending one. The modern proof, the one Lam gives, is a page long and runs entirely through the Jacobson radical.

The theorem is really about modules. Over a semiprimary ring — nilpotent radical, semisimple radical quotient — the three finiteness conditions noetherian, artinian and has a composition series collapse into one. Applying that to $R R$ gives the ring-theoretic corollary.

1939Hopkins and Levitzki, independently

3Conditions shown equivalent

SemiprimaryThe hypothesis

$J^{n} = 0$ What makes the filtration finite

Overview

Two results are packaged here. The first, $(4.14)$ , identifies the semisimple rings among all rings: $R$ is semisimple exactly when $rad R = 0$ and $R$ is left artinian, and in fact the descending chain condition on principal left ideals already suffices. The second, $(4.15)$ , is the Hopkins–Levitzki theorem itself.

The logical route is: a left artinian ring has nilpotent radical by $(4.12)$ and semisimple radical quotient by $(4.14)$ , hence is semiprimary; the radical filtration of any module is then finite with semisimple factors, and a semisimple module satisfying either chain condition is a finite direct sum. Finiteness of the number of layers times finiteness of each layer gives a composition series.

M \supseteq J M \supseteq J^{2} M \supseteq \dots \supseteq J^{n} M = 0, J = rad R, J^{n} = 0,

(4.15)

Each factor $J^{i} M / J^{i + 1} M$ is a module over the semisimple ring $R / J$ .

Learning Objectives

State $(4.14)$ and $(4.15)$ with all hypotheses, including the semiprimary condition.
Prove that a semisimple ring has zero radical using the idempotent decomposition $1 = e + f$ .
Prove that DCC on principal left ideals plus $rad R = 0$ implies semisimplicity.
Run the radical-filtration proof of $(4.15)$ for an arbitrary module.
Deduce that left artinian implies left noetherian and that finitely generated modules over left artinian rings have composition series.
Produce counterexamples for the dropped hypotheses: identity, semiprimarity, and the reverse implication.

Definitions

Semiprimary: $rad R$ is nilpotent and $R / rad R$ is semisimple. Every left artinian ring is semiprimary; the converse is false.
Composition series: A chain $0 = M_{0} ⊊ M_{1} ⊊ \dots ⊊ M_{r} = M$ with each $M_{i + 1} / M_{i}$ simple. Its length $r$ is well defined by Jordan–Hölder.
Noetherian module: ACC on submodules; equivalently every submodule is finitely generated.
Artinian module: DCC on submodules; equivalently every nonempty family of submodules has a minimal member.
Right perfect: By Bass's theorem, equivalent to the DCC on principal left ideals — note the crossing of sides, which is not a typographical error.

All rings have an identity. This is not a stylistic convention here: the corollary DCC implies ACC is false for rings without identity.

Core Concepts

Why a filtration converts one chain condition into the other

Chain conditions are stable under extensions: if $N \subseteq M$ and both $N$ and $M / N$ are noetherian, so is $M$ , and likewise for artinian. So for a module with a finite filtration, both conditions are determined by the factors. The Hopkins–Levitzki argument produces a canonical finite filtration whose factors are modules over a semisimple ring, where the two conditions coincide.

M

artinian or noetherianeach

J^{i} M / J^{i + 1} M

artinian or noetherianeach factor a finite direct sum of simples

M

has a composition series

Semisimple modules are where the two conditions meet

Let $S$ be a semisimple ring and $N$ a left $S$ -module, so $N = ⨁_{i \in I} N_{i}$ with each $N_{i}$ simple. If $I$ is infinite, the partial sums give both a strictly ascending and a strictly descending infinite chain, so $N$ is neither noetherian nor artinian. Hence either chain condition forces $I$ finite, and then $N$ has a composition series of length $| I |$ .

Key Results

Theorem(4.14)Recognising semisimple rings

For a ring $R$ the following are equivalent:

$R$ is semisimple, i.e. $R R$ is a direct sum of simple left modules;
$R$ is J-semisimple and left artinian;
$R$ is J-semisimple and satisfies the DCC on principal left ideals.

By Bass's theorem the ring-theoretic condition in (3) says $R$ is right perfect, so $(1) \Leftrightarrow (3)$ reads: semisimple equals semiprimitive plus right perfect.

Proof

**(1) $\Rightarrow$ (2).** A semisimple ring is left artinian. For the radical, put $𝔄 = rad R$ and use semisimplicity of $R R$ to write $R = 𝔄 \oplus 𝔅$ for some left ideal $𝔅$ . Direct summands of $R R$ are generated by idempotents: $𝔄 = R e$ , $𝔅 = R f$ with $e + f = 1$ and $e, f$ idempotent. Since $e \in rad R$ , the element $f = 1 - e$ is a unit; combined with $f^{2} = f$ this gives $f = 1$ , so $e = 0$ and $𝔄 = 0$ .

**(2) $\Rightarrow$ (3).** Immediate, since principal left ideals are left ideals.

**(3) $\Rightarrow$ (1).** Two consequences of the hypothesis. (a) Every nonzero left ideal $𝔄$ contains a minimal left ideal: choose a minimal member of the nonempty family of nonzero principal left ideals inside $𝔄$ , which exists by the DCC on principal left ideals; such a member is minimal as a left ideal. (b) Every minimal left ideal $𝔅$ is a direct summand of $R R$ : since $𝔅 \neq 0 = rad R$ , there is a maximal left ideal $𝔪$ with $𝔅$ not contained in $𝔪$ ; then $𝔅 \cap 𝔪$ is a proper submodule of the minimal $𝔅$ , hence zero, and $𝔅 + 𝔪 = R$ by maximality, so $R = 𝔅 \oplus 𝔪$ .

Now suppose $R$ is not semisimple. Take a minimal left ideal $𝔅_{1}$ and write $R = 𝔅_{1} \oplus 𝔄_{1}$ using (b). Then $𝔄_{1} \neq 0$ , so by (a) it contains a minimal left ideal $𝔅_{2}$ , which is a summand of $R R$ by (b) and hence of $𝔄_{1}$ ; write $𝔄_{1} = 𝔅_{2} \oplus 𝔄_{2}$ . Iterating produces a strictly descending chain $𝔄_{1} ⊋ 𝔄_{2} ⊋ \dots$ of direct summands of $R R$ . Direct summands are generated by idempotents, hence principal, so this contradicts the DCC on principal left ideals. Therefore the process terminates and $R = 𝔅_{1} \oplus \dots \oplus 𝔅_{r}$ is semisimple.

Theorem(4.15)Hopkins–Levitzki (1939)

Let $R$ be a semiprimary ring: $J : = rad R$ is nilpotent and $R / J$ is semisimple. Then for every left $R$ -module $M$ the following are equivalent:

$M$ is noetherian;
$M$ is artinian;
$M$ has a composition series.

Consequently: (A) a ring is left artinian if and only if it is left noetherian and semiprimary; (B) every finitely generated left module over a left artinian ring has a composition series.

Proof

That (3) implies (1) and (2) is standard: a module with a composition series satisfies both chain conditions, by induction on the length using stability of the chain conditions under extensions.

Conversely assume $M$ is noetherian or artinian. Fix $n$ with $J^{n} = 0$ and consider the finite filtration $M \supseteq J M \supseteq J^{2} M \supseteq \dots \supseteq J^{n} M = 0$ . It suffices to show each factor $J^{i} M / J^{i + 1} M$ has a composition series, since concatenating finitely many finite chains gives one for $M$ .

Each factor is annihilated by $J$ , hence is a module over $\bar{R} = R / J$ , and its $R$ -submodules and $\bar{R}$ -submodules coincide. As a subquotient of $M$ it inherits whichever chain condition $M$ has. But $\bar{R}$ is semisimple, so the factor is a direct sum of simple $\bar{R}$ -modules; a chain condition on that direct sum forces the index set to be finite, so the factor has a composition series of finite length.

For (A): a left artinian ring is semiprimary by $(4.12)$ and $(4.14)$ , so applying the equivalence to $R R$ turns artinian into noetherian; conversely a left noetherian semiprimary ring has $R R$ noetherian, hence artinian. For (B): a finitely generated module over a left artinian ring is a quotient of $R^{m}$ and hence artinian, so it has a composition series by the equivalence.

Corollary—DCC implies ACC for rings

Every left artinian ring with identity is left noetherian. The converse fails: $ℤ$ is noetherian and not artinian, and indeed is not semiprimary since $ℤ / rad ℤ = ℤ$ is not semisimple.

Counterexample—The identity cannot be dropped

Give the Prüfer group $ℤ (p^{\infty})$ the zero multiplication. The resulting ring without identity is artinian — its ideals are the finite cyclic subgroups, totally ordered and satisfying the DCC — but not noetherian, since those subgroups form a strictly ascending infinite chain. Hopkins–Levitzki genuinely uses the identity, through the existence of maximal left ideals and the unit group.

Remark—Left artinian does not imply right artinian

The theorem is one-sided in hypothesis and conclusion. The triangular ring $R = (\begin{matrix} ℚ & ℝ \\ 0 & ℝ \end{matrix})$ is right artinian and right noetherian but neither left artinian nor left noetherian, because $ℝ$ is infinite-dimensional as a $ℚ$ -vector space. So artinian without a side is an unsafe word.

Proof Techniques and Method

The technique, isolated for reuse.

Move 1

Filter by powers of the radical

Whenever the radical is nilpotent, every module carries a canonical finite filtration with factors over $R / rad R$ . Almost every theorem about semiprimary rings is proved by verifying the statement on the factors and lifting.

Move 2

Use extension-stability of chain conditions

Noetherian and artinian are both closed under submodules, quotients and extensions. That is what lets a property of the finitely many factors be reassembled into a property of $M$ .

Move 3

Reduce to semisimple, where finiteness is counting

Over a semisimple ring a module is a direct sum of simples, and every finiteness condition becomes finiteness of the index set. This is the step that makes the two chain conditions coincide.

The same three moves prove the Krull–Schmidt theorem for modules of finite length and the existence of Loewy series, and they reappear in the semiperfect and perfect ring theory of §23–§24, with T-nilpotency substituting for nilpotency.

Worked Example

A uniserial module over a triangular ring

Let $k$ be a division ring, $R = T_{3} (k)$ and $V = k^{3}$ the module of column vectors with the usual matrix action. Let $V_{i} \subseteq V$ be the columns whose last $3 - i$ entries vanish, so $0 = V_{0} ⊊ V_{1} ⊊ V_{2} ⊊ V_{3} = V$ .

Each $V_{i}$ is an $R$ -submodule because upper triangular matrices preserve the flag, and $V_{i} / V_{i - 1}$ is one-dimensional over $k$ with a matrix acting through its $(i, i)$ entry. So $V_{i} / V_{i - 1} ≅ M_{i}$ , the $i$ -th simple module, and

0 ⊊ V_{1} ⊊ V_{2} ⊊ V_{3} = V, V_{i} / V_{i - 1} ≅ M_{i}, J^{i} V = V_{3 - i},

(E.1)

a composition series of length $3$ with pairwise non-isomorphic factors; it is the only one, so $V$ is uniserial.

The radical filtration and the composition series coincide here because each Loewy layer happens to be simple. Hopkins–Levitzki is visible in miniature: $V$ is artinian because it is finite-dimensional, and the theorem correctly predicts that it is noetherian and of finite length.

A module that is artinian and not noetherian

Take $R = ℤ$ and $M = ℤ (p^{\infty})$ , the Prüfer $p$ -group. Its proper submodules are the finite cyclic groups $ℤ / p^{i} ℤ$ , forming a chain

0 ⊊ ℤ / p ℤ ⊊ ℤ / p^{2} ℤ ⊊ \dots ⊊ ℤ (p^{\infty}),

(E.2)

descending chains stabilise, ascending ones need not: $M$ is artinian and not noetherian.

There is no contradiction with $(4.15)$ : $ℤ$ is not semiprimary, since its radical is zero and $ℤ$ is not semisimple. The example shows the hypothesis is doing real work and is not an artefact of the proof.

Comparison and Classification

Chain conditions across standard rings
	Left artinian	Left noetherian	Right artinian	Semiprimary
$M_{n} (D)$	yes	yes	yes	yes
$T_{n} (k)$ , $k$ a field	yes	yes	yes	yes
$ℤ / 72 ℤ$	yes	yes	yes	yes
$ℤ$	no	yes	no	no
$k [x]$	no	yes	no	no
$k [[x]]$	no	yes	no	no
$(\begin{matrix} ℚ & ℝ \\ 0 & ℝ \end{matrix})$	no	no	yes	yes
$k ⟨ x, y ⟩$	no	no	no	no

Chain conditions across standard rings

Which implications hold, and why
Implication	Status	Reason or counterexample
left artinian implies left noetherian	true for rings with identity	$(4.15)$ (A)
left noetherian implies left artinian	false	$ℤ$
artinian implies noetherian for modules	true over semiprimary rings only	$ℤ (p^{\infty})$ over $ℤ$
left artinian implies right artinian	false	the $(ℚ, ℝ)$ triangular ring
artinian implies noetherian without identity	false	$ℤ (p^{\infty})$ with zero multiplication
semiprimary implies left artinian	false	Lam Exercise 20.5

Relationship Map

Semiprimary ring — $J$ nilpotent, $R / J$ semisimple
- for modules
  - noetherian, artinian and finite length coincide
  - every module has a finite radical filtration with semisimple factors
- for the ring itself
  - left artinian is equivalent to left noetherian
  - left artinian is equivalent to the regular module having finite length
- sits between
  - left artinian rings, which are all semiprimary
  - left and right perfect rings, which are all semilocal

Composition seriesNoetherian and artinianFinite length

The equivalence on the right holds for modules over any ring; the content of Hopkins–Levitzki is that over a semiprimary ring either chain condition alone already implies both.

Applications and Industry Use

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

Representation theory

Finite length is the standing hypothesis

Brauer theory, decomposition matrices and block theory all take place in categories of finite-length modules over finite-dimensional algebras. Hopkins–Levitzki is what makes finite length automatic from either chain condition.

Computer algebra

Termination certificates

Module decomposition routines in GAP and Magma assume finite length. For an algebra over a field the assumption is discharged by this theorem rather than checked at runtime.

Coding theory

Modules over finite chain rings

Every finite ring is artinian, hence semiprimary, so codes viewed as modules automatically have finite length and a well-defined composition structure — the basis of rank and type parameters for codes over $ℤ / p^{n}$ .

Algebraic geometry

Artinian local rings

Fat points and infinitesimal neighbourhoods are spectra of artinian local rings; finiteness of length is the numerical invariant — the multiplicity — and Hopkins–Levitzki underwrites its existence.

Computational Notes

Computational notes cover algorithms, cost and library behaviour rather than manufacturing process.

Finite length is what makes module computation possible: the Meataxe and its descendants work with modules of finite length over finite-dimensional algebras, splitting them by finding a submodule or certifying irreducibility.
Hopkins–Levitzki is the guarantee that a finitely generated module over a finite-dimensional algebra has a terminating composition series computation; the number of layers is bounded by $\dim_{k} M$ .
The radical filtration is computed by repeatedly multiplying a basis of $M$ by a basis of $J$ and taking spans; the cost is dominated by the linear algebra, roughly $\dim_{k} J$ matrix-vector products per layer.
Jordan–Hölder guarantees that the multiset of composition factors returned by any correct implementation is independent of the choices made, which is what makes decomposition results comparable across systems.

Failure Modes and Common Mistakes

Do not confuse $(4.14)$ with the definition of semisimplicity; the content is that a chain condition can replace the direct-sum decomposition once the radical vanishes.
Do not assume a finitely generated module over a left noetherian ring has a composition series — that needs artinian, or semiprimary plus noetherian.
Do not read DCC on principal left ideals as a mild weakening; it defines the right perfect rings, a strictly larger class than the left artinian ones.

Historical Notes and Lessons Learned

1908WedderburnStructure theory for finite-dimensional algebras, where both chain conditions hold automatically and the question does not arise.
1927ArtinExtends the structure theory to rings with chain conditions, assuming both ascending and descending throughout, apparently unaware that one implies the other.
1939Hopkins and LevitzkiProve independently that for rings with identity the descending chain condition on left ideals implies the ascending one, removing half of Artin's hypothesis.
1945JacobsonThe general radical supplies the proof now considered standard: filter by powers of the radical and reduce to the semisimple quotient.
1960BassIntroduces perfect rings and shows that DCC on principal left ideals characterises right perfect rings, which is the condition appearing in (4.14)(3).
1968 onwardsRelative versionsNastasescu and later Albu prove relative Hopkins-Levitzki theorems for torsion theories and Grothendieck categories, where the radical filtration is replaced by a localisation-theoretic one.

The lesson repeated across this chapter: a theorem that looks like it is about chain conditions is often really about a filtration. Artin's redundant hypothesis went unnoticed for twelve years because the filtration by powers of the radical was not yet available.

Quick Reference

Hypothesis

R

semiprimary:

J

nilpotent,

R / J

semisimple

Statementnoetherian, artinian and finite length agree for modules

Corollary Aleft artinian equals left noetherian plus semiprimary

Corollary Bf.g. modules over left artinian rings have composition series

(4.14)semisimple equals J-semisimple plus left artinian

(4.14)(3)DCC on principal left ideals, i.e. right perfect

Key devicethe filtration

M \supseteq J M \supseteq \dots \supseteq J^{n} M = 0

Counterexample

ℤ (p^{\infty})

over

ℤ

: artinian, not noetherian

Hypothesis audit
Drop this	What breaks	Witness
semiprimary	artinian no longer implies noetherian	$ℤ (p^{\infty})$ over $ℤ$
identity	DCC no longer implies ACC for rings	$ℤ (p^{\infty})$ with zero product
$J$ nilpotent, keeping $R / J$ semisimple	the filtration becomes infinite	$k [[x]]$ , which is semilocal but not semiprimary
one side	no conclusion on the other side	the $(ℚ, ℝ)$ triangular ring
finite generation in (B)	composition series may not exist	an infinite direct sum of simples over $M_{n} (D)$

Frequently Asked Questions

Why is the theorem stated for modules rather than just for rings?

Because the module version is what gets used and it is no harder to prove. The ring corollary is the case $M =_{R} R$ . Stating it for modules also makes clear that the semiprimary hypothesis is about the base ring, not about the module, which is what allows it to be applied to arbitrary finitely generated modules at once.

Does the converse of Corollary A hold?

Corollary A is already an equivalence: left artinian if and only if left noetherian and semiprimary. What fails is the naive converse left noetherian implies left artinian, and $ℤ$ is the standard counterexample; it is noetherian but not semiprimary.

Is there a version for right modules?

Yes, with the mirror hypothesis. Semiprimarity is left-right symmetric — $rad R$ is symmetric and $R / rad R$ is semisimple on one side exactly when it is on the other — so a semiprimary ring satisfies the theorem for both left and right modules. This is why the ring is left artinian if and only if it is left noetherian, and independently right artinian if and only if right noetherian.

Where exactly is nilpotency of the radical used?

Only to make the filtration finite. If $J$ were merely nil or T-nilpotent, the chain $M \supseteq J M \supseteq \dots$ need not reach zero in finitely many steps and the concatenation of composition series would fail. That is precisely why perfect rings, where $J$ is T-nilpotent, satisfy a weaker theorem.

What is the length of the regular module over a left artinian ring?

It is finite and equals the sum over the radical layers of the number of simple summands in each. For $T_{n} (k)$ over a field it is $n (n + 1) / 2$ ; for $ℤ / p^{n} ℤ$ it is $n$ ; for $M_{n} (D)$ it is $n$ .

Who proved it, Hopkins or Levitzki?

Both, independently, in 1939. Charles Hopkins published in the Annals of Mathematics and Jakob Levitzki in Compositio Mathematica; the convention is to name both. The proof given here is the later radical-theoretic one, not either original argument.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §4, (4.14)–(4.15) (pp. 57–59).
C. Hopkins, “Rings with minimal condition for left ideals”, Annals of Mathematics 40 (1939), 712–730.
J. Levitzki, “On rings which satisfy the minimum condition for the right-hand ideals”, Compositio Mathematica 7 (1939), 214–222.
H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the American Mathematical Society 95 (1960), 466–488.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992, §10 and §15.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964, Chapter I.

AI Suggested Questions

Prove that noetherian and artinian are both closed under extensions of modules, the step assumed in the proof.
Show that a semisimple module satisfying either chain condition is a finite direct sum of simple modules.
Construct a semiprimary ring that is neither left nor right artinian, and compute its radical.
State and prove the relative Hopkins–Levitzki theorem for a hereditary torsion theory.
How does the theorem change for left perfect rings, where the radical is only left T-nilpotent?
Compute the composition length of the regular module for the group algebra of a $p$ -group over $𝔽_{p}$ .

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

Why a filtration converts one chain condition into the other

Semisimple modules are where the two conditions meet

06Key Results

07Proof Techniques and Method

Filter by powers of the radical

Use extension-stability of chain conditions

Reduce to semisimple, where finiteness is counting

08Worked Example

A uniserial module over a triangular ring

A module that is artinian and not noetherian

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

Finite length is the standing hypothesis

Termination certificates

Modules over finite chain rings

Artinian local rings

12Computational Notes

13Failure Modes and Common Mistakes

14Historical Notes and Lessons Learned

15Quick Reference

16Frequently Asked Questions

17Related KEVOS Topics

18References

19AI Suggested Questions

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