CONTENTS
Preface = ⅶ
Recommendations to the reader = ⅸ
Introduction = 1
1. Groups = 1
2. Rings, fields = 19
3. Modules and representations = 32
Chapter 1. Commutative algebra = 51
1. Algebraic and transcendental extensions = 51
2. Galois theory = 60
3. Affine rings = 69
4. Modules over principal ideal rings = 76
5. Algebraic sets = 86
6. Normed fields = 95
Chapter 2. Groups = 105
1. Representations of groups = 105
2. Periodic groups = 114
3. Free groups and graphs = 123
4. Representation of groups by generator and relations = 129
5. Simple groups = 138
6. Topological groups = 144
Chapter 3. Associative rings = 157
1. Radical = 157
2. Classical semisimple rings = 165
3. Structure of noetherian rings = 169
4. Central simple algebras = 175
5. Complete rings of fractions = 190
Chapter 4. Lie algebras = 199
1. Linear Lie algebras = 199
2. Universal enveloping algebra = 208
3. Magnus theory of free groups = 215
4. Lie algebras with triangular decomposition = 224
5. Lie algebras and Lie groups = 235
Chapter 5. Homological algebra = 245
1. Complexes of modules = 245
2. Cohomology of groups = 254
3. Splitting of the radical in a finite-dimensional algebra = 267
4. Brauer group = 273
5. Hopf algebras = 280
Chapter 6. Algebraic groups = 289
1. Hopf algebras and algebraic groups = 289
2. Action of an algebraic group on a set = 305
3. Action of an algebraic group by linear operators = 314
4. Solvable groups = 327
Chapter 7. Varieties of algebras = 335
1. Universal algebras and varieties = 335
2. Finite basis problem for identities in groups = 346
3. PI-algebras = 353
4. Central polynomials for matrix algebras = 364
Set-theoretic supplement = 375
References = 397
Symbol index = 401
Subject index = 405
