Terminology index

2-coboundary, 295 2-cocycle, 295

Absolutely semisimple

algebra, 231

module, 229

Absolutely without radical, module, 246

Adapted basis, 460

Additive

function of modules, 183

set of classes of modules, 183

weakly — function of modules, 183

Adjunction isomorphism, 59

Adverse, 441

Algebra

absolutely semisimple, 231

algebraic, 359

Artinian, 444

Azumaya, 271

central, 251

coinduced, 305

Galois, 310

$(K,G)-$ —, 304

primitive, 443

quaternion, 361

semisimple, 136

similar, 280

simple, 120

© Springer Nature Switzerland AG 2022

N. Bourbaki, Algebra, https://doi.org/10.1007/978-3-031-19293-7

Artinian

algebra, 444

module, 1

ring, 4

Autoinjective

ring, 53

Azumaya algebra, 271

Azumaya’s theorem, 32

Balanced module, 77

Basis, adapted, 460

Bicommutant, 77

Bimodule

invertible, 100

left, 56

Blocks, 180

Brauer group, 279, 330

Burnside’s theorem, 84, 429

Calculus of fractions, 17

Cartan matrix, 201

Cayley (— – Hamilton theorem), 338 Central

algebra, 251

function, 398

Centralizer, 77

Character

Brauer, 434

of a representation, 382, 398

Chevalley (— –Warning theorem), 355 Class of a module, 51

Coboundary, 295

Cocycle, 295

associated with a $\tau$-extension, 296 Coefficient

of a module, 377

of a representation, 377

Cogebra, sub—, 391

Cohomology

group, 324

of algebras, 239

Coinduced

algebra, 305

representation, 403

Colength of a module, 53

Commutant, 77

Compatible homomorphisms, 324

Component, isotypical — of type S, 65 Corestriction, 303, 326

Cotranspose, 338

Countermodule, 8

Cover, projective, 161

Criterion, Mackey’s, 426

Cross product, 315

Decomposable idempotent, 146

Decomposition, Weyr–Fitting, 27

Definition of fractions, 18

Degenerate, non— subalgebra, 370

Degree

left, 124

of a representation, 373, 397

over a subring, 124, 204

reduced — of a simple algebra, 253

Dense subring, 129

Density theorem, 88

Density, Jacobson’s — theorem, 442

Derivation, 9

inner, 240

Description

canonical, 62, 70

of a module, 69

of an isotypical module, 62

Determinant, 452, 453

Diagram, Young, 430

Direct image of a $\tau$-extension, 289

Divisible, cardinal — by, 123

Element

idempotent, 145

integral, 428

pregular, 433

without multiple factors, 155 Equivalence, Morita —

of algebras, 100

of modules, 103

Equivalent idempotents, 40

Essential submodule, 75

Exponent, 322

Extension

of an algebra, 248

product of, 293

splitting field, 281

$\tau$-extension, 285

product of, 293

semitrivial, 286

Factor, simple — of type S, 141

Faithful representation of an algebra, 374 Field

C1, 358

finite, 357

splitting, 281

Field, residue, 26

Fitting, (Weyr– — decomposition), 27 Flat, absolutely — ring, 171

Foot, 130

Form, trace, 464

Formula, Frobenius, 432

Fourier inversion formula, 408

Fractions

calculus of, 17

definition of, 18

monoid of, 17 ring of, 18

Frobenius

formula, 432

reciprocity, 203, 403

ring, 53

Function

additive, of modules, 183

central, 398

weakly additive, of modules, 183

Galois

algebra, 310

group of automorphisms, 274

subring, 274

Gelfand (— –Mazur theorem), 368 Generating module, 79

Goldie’s theorem, 149

Grothendieck

group, 186, 189

ring, 198

Group

Brauer, 279, 330

Galois, of automorphisms, 274

Grothendieck, 186

for direct sums, 189

unimodular, 455, 457

Hamilton (Cayley– — theorem), 338 Height

of a morphism, 204

of a subring, 204

Hereditary set of classes of modules, 183 Hilbert’s

90 theorem, 327

Nullstellensatz, 461

Homomorphism

compatible, 324

corestriction, 303, 326

inflation, 299, 325

linking, 324

of finite rank, 463

restriction, 299, 325

Homothety ring, 1

A VIII.487

Ideal

maximal, 26, 141, 435

minimal, 50

nilpotent, 149, 156

prime, 133

primitive, 133

regular, 435

semiprime two-sided, 149

trace, 80

Idempotent

central, of a group algebra, 408

decomposable, 146

indecomposable, 146

Idempotents, orthogonal, 146

Indecomposable

idempotent, 146

module, 30

Index, 322

of a morphism, 204

of a subring, 204

Induced representation, 402

Inflation homomorphism, 299, 325 Intertwining operator, 398

Inverse image

of a cocycle, 299

of a $\tau$-extension, 287

Inverse of a module, 100

Invertible bimodule, 100

Isomorphism, adjunction, 59

Isotypical

component of type S, 65

module, 61

Jacobson

density theorem, 88

radical, 154

Jacobson’s theorem, 156

Krull–Remak–Schmidt theorem, 37

Lemma

Nakayama’s, 158

Schur’s, 47

Length

of a module, 72

of a ring, 7

Linked modules, 180

Linking homomorphism, 324

Local endomorphism ring, module with

a, 31

Local ring, 26

Mackey’s criterion, 426

Maschke’s theorem, 401

Matrix, Cartan, 201

Maximal

ideal, 26, 141, 435

submodule, 48

Mazur (Gelfand- — theorem), 368

Mean, 325

Minimal, ideal, 50

Module

absolutely semisimple, 229

absolutely without radical, 246

Artinian, 1

balanced, 77

generating, 79

indecomposable, 30

isotypical, 61

Noetherian, 1

of a representation, 373

of finite colength, 53

of type $\mathscr{C}$, 183

primordial, 31

semi-Artinian, 74

semiprimordial, 32

semisimple, 55

simple, 45

with local endomorphism ring, 31

Modules

linked, 180

stably isomorphic, 200

Monoid of fractions, 17

Morita equivalence

of algebras, 100

of modules, 103

Multiplicity, 72, 189

primordial, 34

Multiplier

of a pseudobimodule, 445

of an algebra, 445

Nakayama’s lemma, 158

Nil ideal, 156

Nilpotent ideal, 156

Nilradical, 157

Noether (Skolem– — theorem), 256 Noetherian

module, 1

ring, 4

Norm, reduced, 340

$n$-regular ring, 178

Operator, intertwining, 398

Orthogonal

idempotents, 146

of an ideal, 53

Orthogonality

relation for characters, 410

Schur — relation, 410

second — relation for characters, 412

Partition of an idempotent, 146 Permutation representation, 425 Polynomial

principal, 351

reduced characteristic, 340 Prime

ideal, 133

radical, 167

Primitive

ideal, 133

ring, 120

Primitive algebra, 443

Primordial module, 31

Principal polynomial, 351

Product cross, 315

external tensor, 400

tensor, of representations, 399 Projective cover, 161

Pseudobimodule, 445

Pseudomodule

left, right, 439

semisimple, 442

simple, 440

Pseudoregular ring, 178

Quasi-simple ring, 120 Quaternions, 361

Radical

Jacobson, 154

of a module, 151

of a ring, 154

of an algebra, 440

prime, 167

Reciprocity, Frobenius, 203, 403

Reduced, degree, 253

Regular

ideal, 435

ring, 171

Related representations, 416

Relation

orthogonality, for characters, 410

Schur orthogonality, 410

second orthogonality, for characters,

412

Representation

biregular, 399

coinduced, 403

completely reducible, 374

contragredient, 399

dual, 399

faithful, 374

induced, 402

irreducible, 374

linear

of a group, 397

of an algebra, 373

A VIII.489

matrix, 377

monomial, 427

permutation, 425

quotient, 374

regular, 375, 399

semisimple, 374

simple

of a group, 398

of an algebra, 374

transposed, 374

unit, 399

unitary, 421

Restricted dual, 376

Restriction homomorphism, 299, 325 Ring

absolutely flat, 171

Artinian, 4

autoinjective, 53

Frobenius, 53

Grothendieck, 198

homothety, 1

local, 26

Noetherian, 4

$n$-regular, 178

of fractions, 18

polynomial, 11

primitive, 120

pseudoregular, 178

quasi-simple, 120

semiprime, 149

semisimple, 136

simple, 120

von Neumann regular, 171

without radical, 154

Zorn, 171

Saturation, left, right, 19

Schur orthogonality relation, 410

Schur’s lemma, 47

Second orthogonality relation for characters, 412

Semi-Artinian module, 74

Semiprime ring, 149

Semiprimordial module, 32

Semisimple

algebra, 136

module, 55

pseudomodule, 442

ring, 136

Semisimplification, 191

Semitrivial $\tau$-extension, 286

Set of classes of modules

additive, 183

hereditary, 183

Similar algebras, 280

Simple

algebra, 120

factor of type S, 141

pseudomodule, 440

representation of a group, 398

ring, 120

Simple module, 45

Skolem (— –Noether theorem), 256

Socle, 65, 130

right, left, 130

Splitting field, 281

Subalgebra

commutative maximal, 261

maximal commutative semisimple, 264

maximal étale, 264

nondegenerate, 370

Subcogebra, 391

Submodule

essential, 75

maximal, 48

Sub-pseudomodule, 439

Subrepresentation, 374

Subring

dense, 129

Galois, 274

weakly Galois, 274

Support of a semisimple module, 66, 190

Theorem

Azumaya’s, 32

Burnside’s, 84, 429

Cayley–Hamilton, 338

Chevalley–Warning, 355

Gelfand–Mazur, 368

Goldie, 149

Hilbert’s 90, 327

Hilbert’s Nullstellensatz, 461

Jacobson’s, 156

Jacobson’s density, 88, 442

Krull–Remak–Schmidt, 37

Maschke’s, 401

Skolem–Noether, 256

Tsen’s, 358, 360

Wedderburn’s, 120, 135, 245, 357

Trace

form, 464

ideal, 80

of a representation, 398

reduced, 340

Tracial mapping, 115

Transvection, 457

Tsen’s theorem, 358, 360

Unimodular group, 455 Unit, left, right, 435

Warning (Chevalley– — theorem), 355 Weakly Galois subring, 274

Wedderburn’s theorem, 120, 135, 245,

357

Weyr (— –Fitting decomposition), 27

Young diagram, 430

Zorn ring, 171