Theorem Dependency Map

Q: Is there a single theorem the whole subject depends on?

Closest is Schur's Lemma (3.6): it is quoted inside Wedderburn–Artin, inside the Density Theorem, and throughout the representation theory of §7–§9. But it is also the cheapest result in the book, needing nothing beyond the definition of a simple module. The more accurate answer is that there are two independent roots — Schur on the module side and (4.1) on the ideal side — and the interesting theorems are exactly those that join them.

Q: Why does the Density Theorem not simply replace Wedderburn–Artin?

It nearly does. (11.19) shows a left primitive ring is dense in End(V_k) for a right vector space V over a division ring k, and when V_k is finite the ring is all of End(V_k) ≅ M_n(k), recovering (3.5) for simple artinian rings. What density does not give directly is the *product* decomposition of a general semisimple ring into finitely many simple components, which is where (3.4) and the Jordan–Hölder uniqueness argument are still needed.

Q: Does the graph run the same way for right modules?

Only where each node is side-neutral. rad R (4.3) and semisimplicity (3.7) are, so the Wedderburn–Artin branch dualises intact. The perfect-ring branch does not: (23.20) mixes sides deliberately and (23.22) exhibits a right perfect ring that is not left perfect.

Executive Summary

Noncommutative ring theory is not a list of theorems; it is a directed graph. Four results carry most of the weight: Schur's Lemma $(3.6)$ , the Wedderburn–Artin Theorem $(3.5)$ , the characterisations of the Jacobson radical $(4.1)$ , and the Density Theorem $(11.16)$ . Almost every other named result in this collection is reachable from those four by a short path.

This page records the edges of that graph — what each theorem consumes and what it feeds — so that a reader can see at a glance which hypotheses are inherited and where a chain condition first enters the argument.

4Load-bearing results

2Independent roots

$(4.12)$ Where DCC enters

8Chapters spanned

Overview

Two results in the subject have essentially no prerequisites. Schur's Lemma needs only that a simple module has no proper nonzero submodules. The characterisation of $rad R$ in $(4.1)$ needs only Zorn's Lemma and the correspondence between maximal left ideals and simple modules. Everything else is built on top.

From those roots the graph splits. One branch adds a finiteness hypothesis — semisimplicity, then DCC — and produces Wedderburn–Artin, Hopkins–Levitzki, and eventually the perfect and semiperfect ring theory of §23–§25. The other branch keeps the ring arbitrary and studies its action on modules, producing primitivity, the Density Theorem and the subdirect decomposition theorems.

Read together with Named Theorems Index, which lists the statements, and Ring Class Hierarchy, which orders the ring classes those statements are about. This page supplies the arrows between them.

Learning Objectives

Name the results that carry the most downstream weight and say why.
State the prerequisites of the Wedderburn–Artin Theorem $(3.5)$ exactly.
Trace the path $(4.1) \to (4.12) \to (4.14) \to (4.15)$ and reconstruct each step.
Explain where the Density Theorem $(11.16)$ sits relative to $(3.5)$ and why it is the more general statement.
Separate results needing DCC, results needing only a nil hypothesis, and results needing nothing.
Detect and avoid circular citation when assembling a proof.

Core Concepts

Three kinds of edge

Not all dependencies are alike. It is worth distinguishing them, because they behave differently when a hypothesis is weakened.

Type A

Structural

$B$ quotes $A$ verbatim in its proof. Weakening $A$ breaks $B$ immediately. Example: $(4.15)$ quotes $(4.12)$ for nilpotence of the radical.

Type B

Generalisation

$B$ subsumes $A$ ; $A$ survives as a corollary. Example: the Density Theorem $(11.16)$ specialises to Wedderburn–Artin for simple left artinian rings $(11.19)$ .

Type C

Methodological

$B$ reuses the technique of $A$ without citing it. Example: the proof of $(23.20)$ (4) $\Rightarrow$ (1) reruns the splitting argument from $(4.14)$ .

Where the chain conditions enter

A useful discipline is to mark each theorem with the weakest finiteness hypothesis it needs. The radical results $(4.1)$ – $(4.11)$ need none. $(4.12)$ needs left DCC. $(4.15)$ needs only semiprimary, which is strictly weaker than left artinian. Bass's Theorem P $(23.20)$ replaces DCC on all left ideals by DCC on principal left ideals.

No finitenessNil radicalSemiprimaryLeft artinian

Arrows read as increasing strength: left artinian implies semiprimary implies nil radical.

Sidedness as a dependency

A left-handed theorem may only be quoted inside a left-handed argument. $rad R$ is side-neutral $(4.3)$ , and semisimplicity is side-neutral $(3.7)$ , so those two may be quoted freely. Primitivity and perfectness are not, and quoting them across sides is the commonest way a dependency chain silently fails — see Left–Right Symmetry.

Key Results

The following four statements form the spine. Their hypotheses are given in full because the dependency graph is only as reliable as the hypotheses attached to its nodes.

Lemma(3.6)Schur

Let $R$ be any ring and let $V$ be a simple left $R$ -module. Then $End (_{R} V)$ is a division ring. No chain condition, no algebra structure and no hypothesis on $R$ is required.

Theorem(3.5)Wedderburn–Artin

Let $R$ be a left semisimple ring, that is, a ring whose left regular module $_{R} R$ is a direct sum of simple submodules. Then

R ≅ M_{n_{1}} (D_{1}) \times \dots \times M_{n_{r}} (D_{r})

(3.5)

for division rings $D_{1}, \dots, D_{r}$ and integers $n_{i} \geq 1$ . The integer $r$ , and the pairs $(n_{i}, D_{i})$ up to permutation and isomorphism, are uniquely determined; $R$ has exactly $r$ isomorphism classes of simple left modules.

Rests on: the equivalences for semisimple rings $(2.5)$ ; the structure of $M_{n} (D)$ $(3.3)$ ; closure of semisimplicity under finite products $(3.4)$ ; Schur's Lemma $(3.6)$ ; the identification $End (_{R} R) ≅ R$ with endomorphisms of left modules written on the right; and the Jordan–Hölder Theorem for uniqueness.

Theorem(11.16)Jacobson–Chevalley Density Theorem

Let $V$ be a semisimple left $R$ -module, let $k = End (_{R} V)$ — a division ring when $V$ is simple, by $(3.6)$ — and regard $V$ as a right $k$ -vector space. Then the image of $R$ in $End (V_{k})$ is a dense ring of linear transformations: for every finite $k$ -independent set $v_{1}, \dots, v_{n} \in V$ and every $w_{1}, \dots, w_{n} \in V$ there is $r \in R$ with $r v_{i} = w_{i}$ for all $i$ .

Rests on: the semisimple-module lemma $(11.15)$ and Schur's Lemma. Feeds: the structure theorem for left primitive rings $(11.19)$ , Wedderburn–Artin recovered for simple left artinian rings, and Burnside's Theorem in §9.

Theorem(4.15)Hopkins–Levitzki

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

In particular a ring is left artinian if and only if it is left noetherian and semiprimary, and every finitely generated module over a left artinian ring has a composition series.

The following proof is included not for its own sake but because it exhibits a dependency chain of length four in a single page of argument.

Proof

Claim: every left artinian ring is left noetherian. Let $R$ be left artinian and put $J = rad R$ .

*Step 1 — invoke $(4.12)$ .* Left DCC forces $J$ to be nilpotent: choose $n$ with $J^{n} = 0$ . (The proof of $(4.12)$ itself uses the unit criterion $(4.5)$ , so this step already carries $(4.1)$ with it.)

*Step 2 — invoke $(4.14)$ .* The ring $\bar{R} = R / J$ has zero radical by $(4.6)$ and is left artinian as a quotient of a left artinian ring, hence is semisimple. So $R$ is semiprimary.

Step 3 — filtration. Consider $R \supseteq J \supseteq J^{2} \supseteq \dots \supseteq J^{n} = 0$ . Each factor $J^{i} / J^{i + 1}$ is annihilated by $J$ , so it is a module over $\bar{R}$ .

*Step 4 — invoke $(2.5)$ and $(1.19)$ .* Over the semisimple ring $\bar{R}$ every module is semisimple, so $J^{i} / J^{i + 1}$ is a direct sum of simple modules. It is artinian, being a subquotient of the artinian module $_{R} R$ , and an artinian semisimple module is a finite direct sum of simples; hence it has a composition series and is noetherian. A module with a finite filtration whose factors are all noetherian is noetherian, so $_{R} R$ is noetherian.

Every link is essential: drop Step 1 and the filtration is infinite; drop Step 2 and the factors are not semisimple; drop the artinian hypothesis in Step 4 and a semisimple module can be an infinite direct sum, which is neither noetherian nor artinian.

Corollary(4.14)The hinge

For any ring $R$ the following are equivalent: $R$ is semisimple; $R$ is $J$ -semisimple and left artinian; $R$ is $J$ -semisimple and satisfies DCC on principal left ideals. The third condition is exactly * $J$ -semisimple and right perfect*, which is why $(4.14)$ reappears inside the proof of Bass's Theorem P $(23.20)$ .

Remark—What has no upstream

Schur's Lemma $(3.6)$ , the radical characterisations $(4.1)$ , Brauer's Lemma $(10.22)$ and Fitting's Decomposition Theorem $(19.16)$ each depend on nothing beyond definitions and elementary module theory. They are the safest results to quote and the ones to reach for first when a dependency chain must be shortened.

Proof Techniques and Method

How these proofs work, and which move to reuse.

Four recurring moves account for most of the edges in the graph.

Move 1

Quotient by the radical, then lift

Prove the statement for $R / rad R$ , which is semisimple under a chain condition, then lift using idempotent lifting $(21.28)$ or Nakayama $(4.22)$ . This is the shape of §21–§25 almost throughout.

Move 2

Replace DCC by a minimal counterexample

Assume the conclusion fails, choose a minimal object among those witnessing failure, and derive a contradiction from minimality. Used in $(4.12)$ , in $(4.14)$ (3) $\Rightarrow$ (1), and in Utumi's Lemma $(10.29)$ .

Move 3

Test against a simple module

Convert an ideal-theoretic claim into a module-theoretic one via $𝔪 \leftrightarrow R / 𝔪$ , then apply Schur or density. This is what makes $(4.1)$ , $(11.5)$ and the primitivity theory interlock.

Move 4

Pass to the opposite ring

A theorem proved for right modules transfers to left modules over $R^{op}$ — but only if every hypothesis is itself side-symmetric. Verify each hypothesis before using this.

Move 1 is the reason the radical is introduced at all. It converts a question about an arbitrary ring into a question about a semisimple ring plus a lifting problem, and the lifting problem is exactly what §21–§24 solve.

Frameworks and Models

The eight chapters group into four programmes, each with its own summit theorem.

Programme I — Classify the finite objects — §1–§3, summit $(3.5)$
- Chain conditions and examples (§1)
- Semisimple modules and rings (§2)
- Wedderburn–Artin and simple artinian rings (§3)
Programme II — Measure the failure — §4–§6, §10, summit $(4.15)$ and $(10.27)$
- The Jacobson radical and its behaviour under change of rings (§4–§5)
- Group rings and $J$ -semisimplicity (§6)
- Prime, semiprime and the nilradicals (§10)
Programme III — Act on modules — §7–§9, §11–§12, summit $(11.16)$
- Representation theory of algebras and groups (§7–§8)
- Linear groups (§9)
- Primitive rings and density (§11)
- Subdirect products and commutativity (§12)
Programme IV — Decompose by idempotents — §19–§25, summit $(23.20)$ and $(19.21)$
- Local and semilocal rings (§19–§20)
- Idempotents, corner rings, blocks (§21–§22)
- Perfect and semiperfect rings and their homological characterisations (§23–§24)
- Principal indecomposables and basic rings (§25)
Interlude — Division rings and orderings — §13–§18, largely independent of Programme IV
- Division ring theory and classical constructions (§13–§16)
- Ordered rings and ordered division rings (§17–§18)

Process and Workflow

A practical routine for checking that a proof you are writing is well founded.

List the results you citeWrite down each numbered result the argument invokes, with its side (left or right) and its finiteness hypothesis.

Check hypotheses are met at the point of useA result about

R

is not automatically a result about

R / rad R

, about

M_{n} (R)

, or about a corner ring

e R e

— although for many of these there is a transfer theorem.

Check sidedness is consistentMark each cited result L, R or symmetric. A mixed chain needs an explicit passage through

R^{op}

, and that passage must preserve every hypothesis.

Look for circularityIf your target is used, directly or transitively, in the proof of something you cite, the argument is circular. The commonest trap is proving Wedderburn–Artin from density and then using Wedderburn–Artin inside the density argument.

Shorten to the rootsWhere possible, replace a citation of a downstream theorem by a citation of its root —

(3.6)

(4.1)

— which frees the argument from the hypotheses the downstream theorem carries.

Which theorem should I reach for first?

The ring is left artinian

(4.12)

then

(4.14)

then

(4.15)

. You get a nilpotent radical, a semisimple quotient, and composition series for finitely generated modules — the strongest package available.

The ring is arbitrary but the module is simpleSchur

(3.6)

and the Density Theorem

(11.16)

. Neither needs a chain condition; density is the correct substitute for Wedderburn–Artin in the infinite-dimensional setting.

The ring is arbitrary and the question is about nilpotence

(10.11)

–

(10.14)

for the lower nilradical and

(4.11)

for the containment in

rad R

. Do not expect an answer for nil one-sided ideals: that is Köthe's conjecture.

The question is about lifting decompositions

(21.28)

for idempotents modulo a nil ideal,

(19.21)

for uniqueness of decompositions,

(23.6)

for semiperfect rings.

Comparison and Classification

Prerequisites and consumers of the principal theorems
Theorem	Rests on	Feeds
Schur $(3.6)$	definitions only	$(3.5)$ , $(11.16)$ , §7 splitting fields
Wedderburn–Artin $(3.5)$	$(2.5)$ , $(3.3)$ , $(3.4)$ , $(3.6)$ , Jordan–Hölder	$(3.7)$ , $(3.10)$ , $(8.1)$ , §25
Radical characterisations $(4.1)$	Zorn's Lemma	all of §4–§6, $(11.5)$ , $(21 . x)$
Nil implies radical $(4.11)$	$(4.1)$	$(4.12)$ , $(10.27)$ , $(21.28)$
$rad R$ nilpotent $(4.12)$	left DCC, $(4.5)$	$(4.13)$ , $(4.15)$ , $(10.27)$ , §23
Semisimple criterion $(4.14)$	$(2.5)$ , idempotent splitting	$(4.15)$ , $(23.20)$
Hopkins–Levitzki $(4.15)$	$(4.12)$ , $(4.14)$ , $(1.19)$	§23 semiprimary theory
Nakayama $(4.22)$	$(4.1)$ , $(4.5)$	§19, §21, §24, projective covers
Maschke $(6.1)$	$\| G \|$ invertible in $k$	§8 ordinary representation theory
Density $(11.16)$	$(11.15)$ , $(3.6)$	$(11.19)$ , $(11.20)$ , Burnside §9
Amitsur–McCoy $(10.19)$	$(10.11)$ , $(10.13)$	nilradicals of polynomial rings
Levitzki $(10.30)$	Utumi's Lemma $(10.29)$	Köthe's conjecture for right noetherian rings
Krull–Schmidt–Azumaya $(19.21)$	$(19.16)$ , $(19.17)$ , local endomorphism rings	§23 $(23.8)$ , §25 basic rings
Idempotent lifting $(21.28)$	nil ideal, binomial argument	$(23.19)$ , $(23.20)$ , $(22.8)$
Bass's Theorem P $(23.20)$	$(23.16)$ , $(21.25)$ , $(21.28)$ , $(24.12)$	$(24.18)$ , $(24.25)$
Flat implies projective $(24.25)$	$(23.20)$ , $(24.12)$ , $(24.23)$	homological characterisation of perfectness

What each theorem actually needs
	Chain condition	Identity element	Field or algebra	Side-neutral
Schur $(3.6)$	no	yes	no	yes
Wedderburn–Artin $(3.5)$	partial	yes	no	yes
Radical $(4.1)$	no	yes	no	yes
$(4.12)$	left DCC	yes	no	no
Hopkins–Levitzki $(4.15)$	semiprimary	yes	no	no
Density $(11.16)$	no	yes	no	no
Maschke $(6.1)$	no	yes	yes	yes
Bass $(23.20)$	DCC on principal left ideals	yes	no	no
Krull–Schmidt $(19.21)$	finite decomposition	yes	no	yes

What each theorem actually needs

Relationship Map

The containment picture below is the dependency graph collapsed onto its hypotheses: an inner band inherits every theorem proved for an outer band.

Arbitrary rings

(3.6)

(4.1)

–

(4.11)

(10.22)

(11.16)

(12.3)

(19.16)

Semilocal

R / rad R

semisimple;

(20 . x)

Semiperfectidempotents lift;

(23.6)

(23.10)

(24.12)

One-sided perfect

rad R

T-nilpotent;

(23.20)

(24.18)

(24.25)

Semiprimary

rad R

nilpotent;

(4.15)

One-sided artinian

(4.12)

(4.13)

(11.7)

Semisimple

(3.5)

(3.7)

(4.14)

The radical chain below is the other axis. Each inclusion is proved once and then used everywhere; the strictness of each is the subject of Counterexamples Catalogue.

{Nil}_{*} R

L - rad R

{Nil}^{*} R

rad R

The first inclusion is (10.32), the last is (10.27); both rest on (4.11).

Failure Modes and Common Mistakes

Do not assume that a result proved for $R$ passes to $M_{n} (R)$ ; it does for $rad$ and for ${Nil}_{*}$ $(10.21)$ , but Morita invariance has to be checked case by case.
Do not import a theorem about finite-dimensional algebras into the general ring setting: $(4.20)$ and $(4.19)$ carry cardinality and algebraicity hypotheses that are easy to drop by accident.
Do not treat semisimple and * $J$ -semisimple* as interchangeable. $(4.14)$ says one is the other plus left DCC, and mid-century sources use the words the other way round.

Best Practices

Cite the weakest sufficient result. If Schur's Lemma suffices, do not cite Wedderburn–Artin.
Annotate every citation with its side. Two letters, L or R, in the margin catch most errors.
When a chain condition is used, say which chain and on which side; artinian alone is ambiguous in this subject.
Prefer $(4.15)$ over $(4.12)$ plus ad hoc filtration arguments — the semiprimary hypothesis is the right level of generality and is reused in §23.
Record which of your steps would survive if Köthe's conjecture were false; results in §10 that assume a positive answer must be flagged.

Historical Notes and Lessons Learned

1907–08WedderburnStructure theorem for finite-dimensional algebras over a field, with the radical defined as the largest nilpotent ideal. The dependency graph begins here.
1927ArtinThe structure theory is extended to rings with DCC. Artin assumed both chain conditions, not realising that left DCC implies left ACC.
1939Hopkins and LevitzkiIndependently prove that left DCC implies left ACC, closing the gap Artin left open. The modern proof routes through the Jacobson radical, which did not yet exist.
1945JacobsonThe radical is redefined by its action on simple modules; the Density Theorem is proved. Chain conditions become optional and the graph acquires its second root.
1960BassPerfect rings are introduced and characterised homologically. The chain condition is weakened to DCC on principal left ideals, and flatness enters the subject.
1960s–70sConsolidationKrull–Schmidt–Azumaya, semiperfect ring theory and block decomposition are assembled into the form used here, completing Programme IV.

The methodological lesson is visible in the graph itself. Every time a theorem was reproved from a weaker hypothesis — nilpotence replaced by quasi-regularity, DCC replaced by DCC on principal left ideals, finite dimension replaced by density — the reproof did not merely generalise the statement; it shortened the dependency chain and made whole later chapters possible.

Quick Reference

RootsSchur

(3.6)

; radical characterisations

(4.1)

Bridge

(4.12)

: left artinian implies

rad R

nilpotent

Hinge

(4.14)

: semisimple

=

J

-semisimple

+

left artinian

Summit, finite side

(3.5)

Wedderburn–Artin;

(4.15)

Hopkins–Levitzki

Summit, infinite side

(11.16)

Density;

(11.19)

primitive ring structure

Summit, idempotent side

(23.20)

Bass;

(19.21)

Krull–Schmidt–Azumaya

Weakest useful finitenesssemiprimary for

(4.15)

; DCC on principal left ideals for

(23.20)

Side-neutral nodes

rad R

, semisimplicity, semiperfectness

Genuinely one-sided nodesartinian, noetherian, primitive, perfect

Shortest route to each summit
Target	Route
$(3.5)$ Wedderburn–Artin	$(2.5) \to (3.3) \to (3.4) \to (3.6) \to (3.5)$
$(4.15)$ Hopkins–Levitzki	$(4.1) \to (4.5) \to (4.12) \to (4.14) \to (4.15)$
$(11.19)$ Primitive structure	$(3.6) \to (11.15) \to (11.16) \to (11.19)$
$(23.20)$ Bass	$(21.28) \to (23.16) \to (23.19) \to (23.20)$
$(24.25)$ Flat implies projective	$(23.20) \to (24.12) \to (24.23) \to (24.25)$
$(10.30)$ Levitzki	$(10.11) \to (10.13) \to (10.29) \to (10.30)$

Frequently Asked Questions

Is there a single theorem the whole subject depends on?

Closest is Schur's Lemma $(3.6)$ : it is quoted inside Wedderburn–Artin, inside the Density Theorem, and throughout the representation theory of §7–§9. But it is also the cheapest result in the book, needing nothing beyond the definition of a simple module. The more accurate answer is that there are two independent roots — Schur on the module side and $(4.1)$ on the ideal side — and the interesting theorems are exactly those that join them.

Why does the Density Theorem not simply replace Wedderburn–Artin?

It nearly does. $(11.19)$ shows a left primitive ring is dense in $End (V_{k})$ for a right vector space $V$ over a division ring $k$ , and when $\dim V_{k}$ is finite the ring is all of $End (V_{k}) ≅ M_{n} (k)$ , recovering $(3.5)$ for simple artinian rings. What density does not give directly is the product decomposition of a general semisimple ring into finitely many simple components, which is where $(3.4)$ and the Jordan–Hölder uniqueness argument are still needed.

Which results would survive if Köthe's conjecture were false?

Everything proved in this collection, because Lam never assumes it. The conjecture affects statements about nil one-sided ideals only: $(10.27)$ , $(10.30)$ and Utumi's Lemma $(10.29)$ are all proved unconditionally, and the material that would be strengthened by a positive answer is clearly marked. See Open Problems.

How do I know a dependency is real rather than an artefact of one particular proof?

Test it by looking for an independent proof. Hopkins–Levitzki genuinely depends on the nilpotence of the radical — every known proof filters by powers of $rad R$ . By contrast, $(4.14)$ (2) $\Rightarrow$ (1) is often proved via the perfect-ring condition (3), but Lam notes an argument that avoids (3), so that edge is optional rather than structural.

Where does the subject stop being a single connected graph?

At §13–§18. Division ring theory and the theory of orderings use the radical and Wedderburn–Artin only lightly; §17–§18 are close to independent of the first twelve sections, and their prerequisites are field theory and elementary group theory rather than module theory. That is why they can be read out of order.

Does the graph run the same way for right modules?

Only where each node is side-neutral. $rad R$ $(4.3)$ and semisimplicity $(3.7)$ are, so the Wedderburn–Artin branch dualises intact. The perfect-ring branch does not: $(23.20)$ mixes sides deliberately and $(23.22)$ exhibits a right perfect ring that is not left perfect.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991 — the whole work; the numbering used here is Lam's.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964.
F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics 13, Springer-Verlag, 1992.
L. H. Rowen, Ring Theory, Volumes I and II, Academic Press, 1988.
H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Transactions of the American Mathematical Society 95 (1960), 466–488.
I. N. Herstein, Noncommutative Rings, Carus Mathematical Monographs 15, Mathematical Association of America, 1968.

AI Suggested Questions

Draw the dependency graph of §23 and §24 explicitly, marking which implication in Bass's Theorem P is proved in each section.
Which theorems in Lam's text would need to be reproved if the identity element were not assumed?
Give an independent proof of Hopkins–Levitzki that does not filter by powers of the radical, or explain why none is known.
Trace exactly where the Jordan-Holder theorem is used in the uniqueness half of Wedderburn-Artin.
Compare this dependency structure with the one in Anderson-Fuller, where projective covers come before the radical.
Which of these theorems have been formalised in a proof assistant such as Lean's mathlib, and what were the hard dependencies there?
Identify the results in this collection whose proofs use the axiom of choice essentially rather than incidentally.

01Executive Summary

02Overview

03Learning Objectives

04Core Concepts

Three kinds of edge

Structural

Generalisation

Methodological

Where the chain conditions enter

Sidedness as a dependency

05Key Results

06Proof Techniques and Method

Quotient by the radical, then lift

Replace DCC by a minimal counterexample

Test against a simple module

Pass to the opposite ring

07Frameworks and Models

08Process and Workflow

09Comparison and Classification

10Relationship Map

11Failure Modes and Common Mistakes

12Best Practices

13Historical Notes and Lessons Learned

14Quick Reference

15Frequently Asked Questions

16Related KEVOS Topics

17References

18AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Core Concepts

Key Results

Proof Techniques and Method

Frameworks and Models

Process and Workflow

Comparison and Classification

Relationship Map

Failure Modes and Common Mistakes

Best Practices

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions