Index of Named Theorems in Noncommutative Ring Theory

Q: Why does Lam number lemmas, theorems and examples in a single sequence?

Because the numbering is an address rather than a classification. A single counter per section means every displayed item can be cited unambiguously, and it removes the need to decide in advance whether something deserves the word *Theorem*. The practical consequence is that (4.14) and (4.15) can be an unnamed theorem followed by a famous one.

Q: How many different theorems are called Wedderburn's Theorem?

In this text, four results carry his name: the Wedderburn–Artin classification (3.5), the Little Theorem on finite division rings (13.1), the criterion for a cyclic algebra to be a division ring (14.9), and the factorisation theorem for minimal polynomials over a division ring (16.9). Always cite the number.

Q: Is Nakayama's Lemma really due to Nakayama?

Partly. The commutative case with M an ideal is Krull's; the module-theoretic formulation is due to Azumaya and Nakayama. Lam records that Nakayama himself proposed calling it Krull–Azumaya in the commutative case and Jacobson–Azumaya in the noncommutative one. The customary name has stuck regardless.

Q: Which named results are still open?

One on this list: Köthe's conjecture (10.28), that a ring with no nonzero nil ideal has no nonzero nil one-sided ideal. It is known for right noetherian rings, algebraic algebras and PI-algebras. Several further problems about group rings — zero divisors, units, semisimplicity of kG in characteristic zero — are also open and are collected on *Open Problems*.

Q: What is the difference between the Density Theorem and Wedderburn–Artin?

Density holds with no chain condition and concludes that R approximates End(V_k) on every finite set of vectors. Wedderburn–Artin adds a chain condition, which makes _k V finite and turns the approximation into an equality. Lam's (11.19) makes this precise and gives a second, independent proof of the classification.

Q: Should I memorise the numbers?

Memorise the hypotheses, not the numbers. The numbers are for citation and are easy to look up; the hypotheses are what make an application correct. The final table above lists, for each headline result, the hypothesis that is most often dropped.

Executive Summary

Roughly forty results in this subject carry a name. Some names are attached to several distinct theorems — Wedderburn to four, Amitsur to five, Bass to three, Burnside to three — and several theorems carry more than one name. This page indexes them all against Lam's numbering, records the hypotheses that are most often dropped in citation, and points at the page in this collection that develops each one.

The index is arranged by chapter, since Lam's numbering $(n . m)$ has $n$ the section number, and section numbers determine chapter. A dependency map and a chronology follow.

25Sections indexed

$\approx 45$ Named results

4Theorems called Wedderburn

$(n . m)$ Numbering scheme

Overview

Names in mathematics are addresses, not attributions. Nakayama's Lemma was proved in pieces by Krull, Azumaya and Nakayama, and Nakayama himself suggested it be called Krull–Azumaya or Jacobson–Azumaya. Krull–Schmidt in its usable form is due to Azumaya. Hopkins–Levitzki was proved independently by two people in the same year. Nothing is lost by using the customary names, but a reader who wants the history should not read it off the label.

For the structural picture behind these results see Noncommutative Ring Theory: A Structural Overview; for their logical order see Theorem Dependency Map.

Learning Objectives

Look up any named theorem by chapter and recover its Lam number.
State the headline theorems with complete hypotheses.
Separate the several results sharing a single name.
Identify which named results are used in the proof of which others.
Prove Brauer's Lemma on minimal left ideals from scratch.
Cite results accurately, distinguishing Lam's numbering from other sources'.

Definitions

Lam numbering $(n . m)$: The $m$ -th displayed item of § $n$ . Definitions, theorems, examples and remarks share one counter, so $(4.14)$ and $(4.15)$ may be a theorem and a named theorem in sequence.
Lemma versus Theorem: Lam's usage is functional rather than hierarchical: Schur's Lemma $(3.6)$ and Nakayama's Lemma $(4.22)$ are among the most consequential results in the book.
Named with a bracket: A result written “Theorem (Amitsur)” is attributed but not customarily named after its author; a result written “Amitsur's Theorem” carries the name in ordinary usage.
Semisimple: In this collection: a direct sum of simple modules over itself. In many pre-1970 sources the same word means zero Jacobson radical, which here is called semiprimitive.

Where a theorem is stated below, the hypotheses are complete as given. Where a table row summarises one, the summary names the hypotheses but is not a substitute for the statement.

Key Results

Seven statements in full

Theorem(3.5)Wedderburn–Artin

Let $R$ be a left semisimple ring. Then $R ≅ M_{n_{1}} (D_{1}) \times \dots \times M_{n_{r}} (D_{r})$ for division rings $D_{i}$ and integers $n_{i} \geq 1$ . The number $r$ is uniquely determined and the pairs $(n_{i}, D_{i})$ are unique up to permutation and isomorphism. There are exactly $r$ isomorphism classes of simple left $R$ -modules.

Theorem(4.15)Hopkins–Levitzki

Let $R$ be semiprimary — $rad R$ nilpotent and $R / rad R$ semisimple. Then for any left $R$ -module $M$ , the conditions noetherian, artinian and has a composition series are equivalent.

Lemma(4.22)Nakayama

For a left ideal $J \subseteq R$ the following are equivalent: (1) $J \subseteq rad R$ ; (2) for every finitely generated left $R$ -module $M$ , $J M = M$ implies $M = 0$ ; (3) for left $R$ -modules $N \subseteq M$ with $M / N$ finitely generated, $N + J M = M$ implies $N = M$ .

Theorem(6.1)Maschke

Let $k$ be any ring and $G$ a finite group. Then $k G$ is semisimple if and only if $k$ is semisimple and $| G | \cdot 1$ is a unit of $k$ .

Theorem(11.16)Density Theorem (Jacobson, Chevalley)

Let $R$ be a ring and $V$ a semisimple left $R$ -module; put $k = End (_{R} V)$ . Then $R$ acts densely on $V_{k}$ : given $f \in End (V_{k})$ and $v_{1}, \dots, v_{n} \in V$ , there is $r \in R$ with $r v_{i} = f (v_{i})$ for every $i$ .

Theorem(19.21)Krull–Schmidt–Azumaya

Let $R$ be a ring and let a right $R$ -module $M$ have two finite decompositions $M = M_{1} \oplus \dots \oplus M_{r} = N_{1} \oplus \dots \oplus N_{s}$ in which every $N_{j}$ is indecomposable and every $M_{i}$ is strongly indecomposable, that is, has a local endomorphism ring. Then $r = s$ and, after reindexing, $M_{i} ≅ N_{i}$ for all $i$ .

Theorem(13.17)Cartan–Brauer–Hua

Let $D$ be a division ring and $K$ a division subring that is invariant under every inner automorphism of $D$ , that is, $a K a^{- 1} \subseteq K$ for all $a \in D^{*}$ . If $K \neq D$ then $K \subseteq Z (D)$ .

One proof in full

Lemma(10.22)Brauer

Let $𝔄$ be a minimal left ideal in a ring $R$ . Then either $𝔄^{2} = 0$ , or $𝔄 = R e$ for some idempotent $e \in 𝔄$ .

Proof

Assume $𝔄^{2} \neq 0$ . Then $𝔄 a \neq 0$ for some $a \in 𝔄$ , and in particular $a \neq 0$ . Now $𝔄 a$ is a left ideal contained in $𝔄$ , so minimality gives $𝔄 a = 𝔄$ . Since $a \in 𝔄 = 𝔄 a$ , there is $e \in 𝔄$ with $e a = a$ .

Put $𝔅 = {x \in 𝔄 : x a = 0}$ . This is a left ideal contained in $𝔄$ , and it is proper because $e \in 𝔄$ satisfies $e a = a \neq 0$ , so $e \notin 𝔅$ . Minimality of $𝔄$ therefore forces $𝔅 = 0$ .

Now $e^{2} a = e (e a) = e a = a$ , so $(e^{2} - e) a = 0$ with $e^{2} - e \in 𝔄$ ; hence $e^{2} - e \in 𝔅 = 0$ and $e$ is idempotent. It is nonzero because $e a = a \neq 0$ . Finally $R e$ is a nonzero left ideal contained in $𝔄$ — nonzero since $e = e \cdot e \in R e$ — so minimality gives $R e = 𝔄$ .

Proof Techniques and Method

How to use this index, and how these results are conventionally cited.

Citing

Number then name

Write “Hopkins–Levitzki $(4.15)$ ” rather than a bare name. Lam's numbers are stable across printings and unambiguous; the names are neither.

Checking

Recover the hypotheses first

Before applying a named result, restate its hypotheses in the notation of your problem. Most misapplications in this subject are dropped side conditions, not logical errors.

Disambiguating

Two theorems, one name

Jacobson's Theorem is $(12.10)$ or $(13.11)$ ; Wedderburn's Theorem is $(14.9)$ or $(16.9)$ ; Bass's Theorem is $(20.9)$ , $(23.20)$ or $(24.25)$ . Always give the number.

Translating

Other sources number differently

Jacobson, Herstein, Anderson–Fuller and Rowen each use their own scheme. When quoting across sources, give the statement, not only the label.

Searching

Search by hypothesis

If you know the hypothesis — semiprimary, semilocal, right noetherian — the tables above narrow the candidates faster than searching by conclusion.

History

Names lag proofs

Nakayama, Krull–Schmidt and Hopkins–Levitzki each acquired their names well after the underlying results appeared, and in at least one case against the wishes of the person named.

Frameworks and Models

Grouping by name rather than by chapter exposes the recurring authors and shows how one person's programme spans the book.

Wedderburn — Four distinct results
- $(3.5)$ Wedderburn–Artin: classification of semisimple rings
- $(13.1)$ Little Theorem: finite division rings are fields
- $(14.9)$ Criterion for a cyclic algebra to be a division ring
- $(16.9)$ Factorisation of minimal polynomials over a division ring
Jacobson — The radical and its consequences
- $(11.16)$ Density Theorem, with Chevalley
- $(12.9)$ Jacobson–Herstein commutativity criterion
- $(12.10)$ $a^{n (a)} = a$ forces commutativity
- $(13.11)$ Algebraic division algebras over finite fields
- $(16.14)$ Niven–Jacobson on quaternionic roots
Amitsur — Radicals under change of rings
- $(4.20)$ Radical of an algebra of small dimension
- $(5.10)$ $rad R [T] = N [T]$
- $(6.12)$ Group algebras over nonalgebraic extensions
- $(10.19)$ Amitsur–McCoy on the lower nilradical
Bass — Perfect rings and stable range
- $(20.9)$ Units in cosets over semilocal rings
- $(23.20)$ Theorem P: right perfect rings
- $(24.25)$ Flat implies projective
Burnside — Linear groups and irreducible actions
- $(7.3)$ Burnside's Lemma on irreducible subalgebras
- $(9.4)$ – $(9.5)$ Two finiteness criteria for linear groups
Herstein — Commutators in division rings
- $(12.9)$ with Jacobson; $(12.11)$ with Kaplansky
- $(13.8)$ Herstein's Lemma
- $(13.26)$ Conjugates of a noncentral element
Levitzki — Nilness under chain conditions
- $(4.15)$ Hopkins–Levitzki
- $(10.30)$ Nil one-sided ideals in right noetherian rings

Comparison and Classification

Chapters I–II: semisimplicity and the radical, §1–§6

Named results, §1–§6
Named result	Lam	Hypotheses and conclusion in brief
Wedderburn–Artin Theorem	$(3.5)$	$R$ left semisimple $\Rightarrow$ finite product of $M_{n_{i}} (D_{i})$ , with uniqueness
Schur's Lemma	$(3.6)$	$V$ a simple left $R$ -module $\Rightarrow$ $End (_{R} V)$ is a division ring
Amitsur's simplicity corollary	$(3.16)$	$k$ simple of characteristic $0$ and $δ$ a non-inner derivation $\Rightarrow$ $k [x; δ]$ is a simple ring that is not artinian
Hopkins–Levitzki Theorem	$(4.15)$	$R$ semiprimary $\Rightarrow$ noetherian, artinian and finite length agree for all modules
Amitsur's Theorem on algebras	$(4.20)$	$R$ a $k$ -algebra with $\dim_{k} R < \| k \|$ as cardinals $\Rightarrow$ $rad R$ is the largest nil ideal of $R$
Nakayama's Lemma	$(4.22)$	$J \subseteq rad R$ iff $J M = M$ forces $M = 0$ for finitely generated $M$
Snapper's Theorem	$(5.1)$	$R$ commutative $\Rightarrow$ $rad R [T] = Nil (R [T]) = (Nil R) [T]$
Amitsur's Theorem on $R [T]$	$(5.10)$	$rad R [T] = N [T]$ with $N = R \cap rad R [T]$ a nil ideal of $R$
Maschke's Theorem	$(6.1)$	$G$ finite $\Rightarrow$ $k G$ semisimple iff $k$ semisimple and $\| G \|$ invertible in $k$
Rickart–Amitsur argument	$(6.5)$ – $(6.6)$	Algebraic and analytic halves of the proof that $ℂ G$ is Jacobson semisimple for every group $G$
Amitsur's Theorem on field extensions	$(6.12)$	$K$ a nonalgebraic field extension of $ℚ$ $\Rightarrow$ $K G$ is Jacobson semisimple for every group $G$

Chapter III: representation theory, §7–§9

Named results, §7–§9
Named result	Lam	Hypotheses and conclusion in brief
Burnside's Lemma	$(7.3)$	$M$ a finite-dimensional right $k$ -vector space, $A$ a $k$ -subalgebra of $End (M_{k})$ with $M$ simple as a left $A$ -module and $End (_{A} M) = k$ $\Rightarrow$ $A = End (M_{k})$
Brauer's Theorem	$(8.9)$	$G$ finite, $k$ a splitting field of characteristic $p > 0$ $\Rightarrow$ the number of irreducible $k G$ -representations equals the number of $p$ -regular classes
Orthogonality Relations	$(8.16)$	First and second orthogonality relations for the irreducible characters of a finite group
Trace Lemma	$(9.3)$	$G$ a subsemigroup of $G L_{n} (k)$ with $k^{n}$ absolutely irreducible over $k G$ , and $\| tr (G) \| = r$ $\Rightarrow$ $\| G \| \leq r^{n^{2}}$
Burnside's First Theorem	$(9.4)$	$k$ of characteristic $p > 0$ and $G \subseteq G L_{n} (k)$ of finite exponent $N$ with $N$ prime to $p$ $\Rightarrow$ $\| G \| \leq N^{n^{3}}$
Burnside's Second Theorem	$(9.5)$	A linear group $G \subseteq G L_{n} (k)$ is finite if and only if it has finitely many conjugacy classes
Schur's Theorem	$(9.9)$	A finitely generated torsion linear group over a field is finite
Lie–Kolchin–Suprunenko Theorem	$(9.18)$	A $λ$ -potent subgroup of $G L (V)$ has all finite-dimensional irreducible modules of dimension one, and is triangularisable in a suitable basis

Chapter IV: prime and primitive rings, §10–§12

Named results, §10–§12
Named result	Lam	Hypotheses and conclusion in brief
Amitsur–McCoy Theorem	$(10.19)$	${Nil}_{} (R [T]) = ({Nil}_{} R) [T]$ for every ring $R$
Brauer's Lemma	$(10.22)$	$𝔄$ a minimal left ideal $\Rightarrow$ $𝔄^{2} = 0$ or $𝔄 = R e$ with $e$ idempotent
Köthe's Conjecture	$(10.28)$	${Nil}^{*} R = 0$ should imply no nonzero nil one-sided ideal — open
Levitzki's Theorem	$(10.30)$	$R$ right noetherian $\Rightarrow$ every nil one-sided ideal is nilpotent and ${Nil}_{} R = {Nil}^{} R$
Density Theorem	$(11.16)$	$V$ a semisimple left $R$ -module $\Rightarrow$ $R$ acts densely on $V$ over $End (_{R} V)$
Structure Theorem for Left Primitive Rings	$(11.19)$	$R$ left primitive with faithful simple $V$ $\Rightarrow$ $R$ is a dense ring of linear transformations on $V_{k}$ ; if left artinian then $R ≅ M_{n} (k)$
Formanek's Theorem	$(11.27)$	$k$ a countable domain, at least two free indeterminates $\Rightarrow$ the free ring over $k$ is left primitive
Jacobson–Herstein Theorem	$(12.9)$	$R$ commutative iff for all $a, b$ there is $n = n (a, b) > 1$ with $(a b - b a)^{n} = a b - b a$
Jacobson's Theorem	$(12.10)$	If for every $a \in R$ there is $n (a) > 1$ with $a^{n (a)} = a$ , then $R$ is commutative
Herstein–Kaplansky Theorem	$(12.11)$	$R$ semiprimitive: commutative iff all additive commutators are central iff every $a$ has some power in the centre

Chapters V–VI: division rings and ordered structures, §13–§18

Named results, §13–§18
Named result	Lam	Hypotheses and conclusion in brief
Wedderburn's Little Theorem	$(13.1)$	Every finite division ring is a field
Herstein's Lemma	$(13.8)$	$D$ of characteristic $p > 0$ and $a$ a noncentral torsion element of $D^{}$ $\Rightarrow$ there is $y \in D^{}$ , an additive commutator, with $y a y^{- 1} = a^{p^{i}} \neq a$ for some $i > 0$
Jacobson's Theorem	$(13.11)$	An algebraic division algebra over a finite field is commutative
Frobenius' Theorem	$(13.12)$	An algebraic division algebra over $ℝ$ is $ℝ$ , $ℂ$ or the real quaternions $ℍ$
Cartan–Brauer–Hua Theorem	$(13.17)$	A division subring invariant under all inner automorphisms is either everything or central
Herstein's Theorem	$(13.26)$	A noncentral element of a division ring has infinitely many conjugates
Wedderburn's Theorem on cyclic algebras	$(14.9)$	If the class of $a$ in $F^{}$ modulo the norms from $K^{}$ has order equal to the degree, the cyclic algebra $(K / F, σ, a)$ is a division $F$ -algebra
Brauer–Albert Theorem	$(15.16)$	$D$ of dimension $r^{2}$ over its centre $F$ $\Rightarrow$ there are $α, β \in D$ such that the $r^{2}$ products $α^{i} β^{j}$ , $0 \leq i, j < r$ , form an $F$ -basis of $D$
Gordon–Motzkin Theorem	$(16.4)$ , $(16.11)$	A polynomial of degree $n$ over a division ring has roots in at most $n$ conjugacy classes
Wedderburn's Factorisation Theorem	$(16.9)$	$A$ a conjugacy class of $D$ algebraic over the centre $F$ with minimal polynomial $f$ of degree $n$ $\Rightarrow$ $f = (t - a_{n}) \dots (t - a_{1})$ in $D [t]$ with all $a_{i} \in A$ , and $a_{1}$ may be prescribed
Bray–Whaples Theorem	$(16.13)$	Reconstruction of a polynomial over a division ring from roots in distinct conjugacy classes
Niven–Jacobson Theorem	$(16.14)$	Over a real-closed field, solvability of polynomial equations in the associated quaternion division algebra
Baer's Theorem	$(16.15)$	$D$ noncommutative and centrally finite with centre $R$ , every polynomial in $R [t]$ having a root in $D$ $\Rightarrow$ $R$ is real-closed and $D$ is the quaternion algebra over $R$
R. E. Johnson's Theorem	$(17.11)$	For $R \neq 0$ : $R$ is formally real $\Leftrightarrow$ $R$ has a preordering $\Leftrightarrow$ $R$ has an ordering
Albert–Neumann–Fuchs Theorem	$(17.17)$	$R \subseteq R'$ domains with $R'$ a ring of quotients of $R$ $\Rightarrow$ every ordering of $R$ extends uniquely to an ordering of $R'$
Szele–Pickert Theorem	$(18.2)$	$D$ can be ordered if and only if $- 1$ is not a sum of square-products in $D$
Scharlau–Tschimmel Theorem	$(18.9)$	Computes the level of a division ring: under the stated hypothesis on the base field, the shortest expression of $- 1$ as a sum of square-products has length $r + 1$

Chapters VII–VIII: idempotents, perfect and semiperfect rings, §19–§25

Named results, §19–§25
Named result	Lam	Hypotheses and conclusion in brief
Fitting Decomposition Theorem	$(19.16)$	$M_{R}$ of finite length and $f$ an endomorphism $\Rightarrow$ $M = \ker (f^{n}) \oplus im (f^{n})$ for all sufficiently large $n$
Krull–Schmidt–Azumaya Theorem	$(19.21)$	Two finite decompositions, one into indecomposables and one into strongly indecomposables, agree up to reindexing and isomorphism
Bass' Theorem on semilocal rings	$(20.9)$	$R$ semilocal, $R a + 𝔅 = R$ $\Rightarrow$ the coset $a + 𝔅$ contains a unit
Cancellation Theorem	$(20.11)$	$End (A_{R})$ of left stable range one $\Rightarrow$ $A \oplus B ≅ A \oplus C$ implies $B ≅ C$
Akizuki–Cohen Corollary	$(23.12)$	A commutative ring is artinian iff it is a finite product of artinian local rings
Bass's Theorem P	$(23.20)$	$R$ right perfect iff DCC on principal left ideals iff every nonzero left module has a simple submodule and no infinite orthogonal idempotent set
Bass: flat implies projective	$(24.25)$	$R$ right perfect iff every flat right $R$ -module is projective

Where to read more in this collection
Named result	Collection page
Wedderburn–Artin Theorem	The Wedderburn–Artin Theorem
Schur's Lemma	Schur's Lemma
Hopkins–Levitzki Theorem	The Hopkins–Levitzki Theorem
Nakayama's Lemma	Nakayama's Lemma
Maschke's Theorem	Maschke's Theorem
Amitsur's Radical Theorem	Amitsur's Radical Theorem
Amitsur–McCoy Theorem	Radical of R[T]
Levitzki's Theorem	The Levitzki Radical
Köthe's Conjecture	Upper Nilradical and Köthe's Conjecture
Density Theorem	The Density Theorem
Jacobson–Herstein Theorem	Commutativity Theorems
Wedderburn's Little Theorem	Wedderburn's Little Theorem
Cartan–Brauer–Hua Theorem	The Cartan–Brauer–Hua Theorem
Gordon–Motzkin Theorem	The Gordon–Motzkin Theorem
Bray–Whaples Theorem	The Bray–Whaples Theorem
Niven–Jacobson Theorem	The Niven–Jacobson Theorem
Krull–Schmidt–Azumaya Theorem	The Krull–Schmidt Theorem
Bass's Theorem P	Bass's Theorem P
Bass: flat implies projective	Flat Implies Projective

Relationship Map

The dependency structure is shallow but definite: a small number of results are used everywhere, and almost nothing depends on the specialised theorems.

Schur

(3.6)

Wedderburn–Artin

(3.5)

Radical theory §4Hopkins–Levitzki

(4.15)

Perfect rings §23

Density

(11.16)

Structure of primitive rings

(11.19)

Wedderburn–Artin recoveredCommutativity theorems

(12.9)

Most heavily used results — If you learn five, learn these
- Schur's Lemma $(3.6)$
  - Used in every structure theorem
  - Supplies the division ring in Wedderburn–Artin and in density
- Nakayama's Lemma $(4.22)$
  - Used in idempotent lifting, projective covers, Krull–Schmidt arguments
- Wedderburn–Artin $(3.5)$
  - Used wherever $R / rad R$ is analysed
  - Underlies the semilocal and semiperfect theory
- Hopkins–Levitzki $(4.15)$
  - Used to convert DCC into finite length
  - Needed for Jordan–Hölder multiplicities over artinian rings
- Density $(11.16)$
  - Used when no chain condition is available
  - Generalises Wedderburn–Artin via $(11.19)$

Theorem Dependency Map gives the full graph, including which results Lam proves twice by independent routes.

Standards and Notation

Standards here covers notation, symbol and markup standards, and reference implementations, rather than material or design codes.

Numbering

(n . m)

= item

m

of §

n

in Lam

Chapter map§1–3 Ch. I; §4–6 Ch. II; §7–9 Ch. III; §10–12 Ch. IV; §13–16 Ch. V; §17–18 Ch. VI; §19–22 Ch. VII; §23–25 Ch. VIII

Ambiguous namesJacobson's, Wedderburn's, Bass's, Amitsur's, Burnside's, Herstein's Theorems each denote more than one result

Alternative namesNakayama

=

Krull–Azumaya

=

Jacobson–Azumaya; Krull–Schmidt

=

Krull–Remak–Schmidt–Azumaya

Terminology driftSemisimple meant semiprimitive in most sources before 1970

MarkupPresentation MathML per ISO/IEC 40314; symbol conventions per ISO 80000-2

Failure Modes and Common Mistakes

Do not cite Maschke's Theorem for infinite groups: the hypothesis that $G$ is finite is essential, and the infinite case is the whole subject of §6.
Do not cite Levitzki's Theorem for left noetherian rings without checking the side; Lam states it for right noetherian rings, and the proof uses ACC on right annihilators.
Do not treat Köthe's statement $(10.28)$ as a theorem. It is a conjecture, open since 1930, and several published arguments have quietly assumed it.

Historical Notes and Lessons Learned

1905Wedderburn's Little TheoremFinite division rings are commutative. The result that starts the subject and is still reproved by new methods every decade.
1907–1909Wedderburn's structure theoryFinite-dimensional algebras decompose as matrix algebras over division algebras modulo the nilpotent radical.
1927ArtinThe structure theory is rebuilt on the descending chain condition, producing what is now called Wedderburn–Artin.
1939Hopkins and LevitzkiIndependently prove that a one-sided artinian ring is noetherian on the same side.
1945JacobsonThe radical is redefined for arbitrary rings; the Density Theorem, with Chevalley, replaces classification when chain conditions are absent.
1950AzumayaKrull–Schmidt uniqueness is proved in the form that actually gets used: local endomorphism rings rather than chain conditions.
1955–1960Amitsur, Herstein, KaplanskyRadicals under change of rings, commutativity theorems, and the structure theory of division rings reach their modern shape.
1960BassPerfect and semiperfect rings; Theorem P and the homological characterisation of perfectness.

One pattern is worth extracting. The results that acquired names are the ones that converted a hypothesis into a structure — chain condition into decomposition, nilness into invertibility, irreducibility into density. Results that merely computed something did not acquire names, however useful they are.

Quick Reference

ClassificationWedderburn–Artin

(3.5)

EndomorphismsSchur

(3.6)

Chain conditionsHopkins–Levitzki

(4.15)

Radical calculusNakayama

(4.22)

Group algebrasMaschke

(6.1)

No chain conditionDensity

(11.16)

, Structure

(11.19)

NilnessLevitzki

(10.30)

, Köthe

(10.28)

Division ringsWedderburn's Little

(13.1)

, Cartan–Brauer–Hua

(13.17)

, Frobenius

(13.12)

DecompositionsFitting

(19.16)

, Krull–Schmidt–Azumaya

(19.21)

PerfectnessBass

(23.20)

(24.25)

The hypothesis most often forgotten
Result	Do not forget
Wedderburn–Artin $(3.5)$	$R$ semisimple, not merely artinian
Hopkins–Levitzki $(4.15)$	$R$ semiprimary
Nakayama $(4.22)$	$M$ finitely generated
Maschke $(6.1)$	$G$ finite and $\| G \|$ invertible in $k$
Density $(11.16)$	$V$ semisimple, and only finitely many vectors matched
Krull–Schmidt–Azumaya $(19.21)$	One side strongly indecomposable
Bass's Theorem P $(23.20)$	Right perfect corresponds to DCC on principal left ideals
Levitzki $(10.30)$	Right noetherian

Frequently Asked Questions

Why does Lam number lemmas, theorems and examples in a single sequence?

Because the numbering is an address rather than a classification. A single counter per section means every displayed item can be cited unambiguously, and it removes the need to decide in advance whether something deserves the word Theorem. The practical consequence is that $(4.14)$ and $(4.15)$ can be an unnamed theorem followed by a famous one.

How many different theorems are called Wedderburn's Theorem?

In this text, four results carry his name: the Wedderburn–Artin classification $(3.5)$ , the Little Theorem on finite division rings $(13.1)$ , the criterion for a cyclic algebra to be a division ring $(14.9)$ , and the factorisation theorem for minimal polynomials over a division ring $(16.9)$ . Always cite the number.

Is Nakayama's Lemma really due to Nakayama?

Partly. The commutative case with $M$ an ideal is Krull's; the module-theoretic formulation is due to Azumaya and Nakayama. Lam records that Nakayama himself proposed calling it Krull–Azumaya in the commutative case and Jacobson–Azumaya in the noncommutative one. The customary name has stuck regardless.

Which named results are still open?

One on this list: Köthe's conjecture $(10.28)$ , that a ring with no nonzero nil ideal has no nonzero nil one-sided ideal. It is known for right noetherian rings, algebraic algebras and PI-algebras. Several further problems about group rings — zero divisors, units, semisimplicity of $k G$ in characteristic zero — are also open and are collected on Open Problems.

What is the difference between the Density Theorem and Wedderburn–Artin?

Density holds with no chain condition and concludes that $R$ approximates $End (V_{k})$ on every finite set of vectors. Wedderburn–Artin adds a chain condition, which makes $\dim_{k} V$ finite and turns the approximation into an equality. Lam's $(11.19)$ makes this precise and gives a second, independent proof of the classification.

Should I memorise the numbers?

Memorise the hypotheses, not the numbers. The numbers are for citation and are easy to look up; the hypotheses are what make an application correct. The final table above lists, for each headline result, the hypothesis that is most often dropped.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991 — the whole work; all numbering above is Lam's.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37, revised edition, 1964.
I. N. Herstein, Noncommutative Rings, Carus Mathematical Monographs 15, Mathematical Association of America, 1968.
N. J. Divinsky, Rings and Radicals, Mathematical Expositions 14, University of Toronto Press, 1965 — for the history of the radical theorems.
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.

AI Suggested Questions

Draw the full dependency graph of the named theorems in Lam and identify which results are proved twice by independent methods.
Which of these theorems have purely categorical proofs, and which genuinely need elements?
Give the modern statement of Bass's Theorem P in terms of projective covers and explain why the sides swap.
Trace the history of the Krull–Schmidt theorem from Wedderburn in 1909 to Azumaya in 1950.
Which of the commutativity theorems of §12 and §13 have quantitative refinements?
How would the index change if one added the results of Lam's companion volume on modules and rings?
Identify the named results whose proofs use the axiom of choice essentially.

Named Theorems Index

Executive Summary

Overview

Learning Objectives

Definitions

Key Results

Seven statements in full

One proof in full

Proof Techniques and Method

Number then name

Recover the hypotheses first

Two theorems, one name

Other sources number differently

Search by hypothesis

Names lag proofs

Frameworks and Models

Comparison and Classification

Chapters I–II: semisimplicity and the radical, §1–§6

Chapter III: representation theory, §7–§9

Chapter IV: prime and primitive rings, §10–§12

Chapters V–VI: division rings and ordered structures, §13–§18

Chapters VII–VIII: idempotents, perfect and semiperfect rings, §19–§25

Relationship Map

Standards and Notation

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Key Results

Seven statements in full

One proof in full

06Proof Techniques and Method

Number then name

Recover the hypotheses first

Two theorems, one name

Other sources number differently

Search by hypothesis

Names lag proofs

07Frameworks and Models

08Comparison and Classification

Chapters I–II: semisimplicity and the radical, §1–§6

Chapter III: representation theory, §7–§9

Chapter IV: prime and primitive rings, §10–§12

Chapters V–VI: division rings and ordered structures, §13–§18

Chapters VII–VIII: idempotents, perfect and semiperfect rings, §19–§25

09Relationship Map

10Standards and Notation

11Failure Modes and Common Mistakes

12Historical Notes and Lessons Learned

13Quick Reference

14Frequently Asked Questions

15Related KEVOS Topics

16References

17AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Key Results

Proof Techniques and Method

Frameworks and Models

Comparison and Classification

Relationship Map

Standards and Notation

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions