Algebra Volume 3, 2nd Edition

Contents......Page 3
Conventions on terminology and notes to the reader......Page 8
1.1 Algebras and homomorphisms......Page 10
1.2 Congruences and the isomorphism theorems......Page 13
1.3 Free algebras and varieties......Page 20
1.4 Abstract dependence relations......Page 29
1.5 The diamond lemma......Page 34
1.6 Ultraproducts......Page 36
1.7 The natural numbers......Page 40
Further exercises......Page 47
2.1 Graded algebras......Page 49
2.2 Free algebras and tensor algebras......Page 52
2.3 The Hilbert series of a graded algebra or module......Page 61
2.4 The exterior algebra on a module......Page 67
Further exercises......Page 75
3.1 Additive and abelian categories......Page 78
3.2 Functors on abelian categories......Page 87
3.3 The category M_R......Page 96
3.4 Homological dimension......Page 103
3.5 Derived functors......Page 108
3.6 Ext and Tor and the global dimension of a ring......Page 119
Further exercises......Page 129
4.1 Group extensions......Page 133
4.2 The Frattini subgroup and the Fitting subgroup......Page 144
4.3 Hall subgroups......Page 150
4.4 The transfer......Page 155
4.5 Free groups......Page 157
4.6 Commutators......Page 163
4.7 Linear groups......Page 168
Further exercises......Page 174
5.1 Algebraic dependence......Page 177
5.2 Simple transcendental extensions......Page 180
5.3 Separable and p-radical extensions......Page 185
5.4 Derivations......Page 190
5.5 Linearly disjoint extensions......Page 195
5.6 Infinite algebraic extensions......Page 205
5.7 Galois cohomology......Page 211
5.8 Kummer extensions......Page 215
Further exercises......Page 219
6.1 The Krull-Schmidt theorem......Page 222
6.2 The projective cover of a module......Page 227
6.3 Semiperfect rings......Page 230
6.4 Equivalence of module categories......Page 236
6.5 The Morita context......Page 243
6.6 Projective, injective and flat modules......Page 248
6.7 Hochschild cohomology and separable algebras......Page 257
Further exercises......Page 265
7.1 Simple Artinian rings......Page 267
7.2 The Brauer group......Page 275
7.3 The reduced norm and trace......Page 282
7.4 Quaternion algebras......Page 289
7.5 Crossed products......Page 292
7.6 Change of base field......Page 298
7.7 Cyclic algebras......Page 304
Further exercises......Page 307
8.1 The Clifford algebra of a quadratic space......Page 309
8.2 The spinor norm......Page 315
8.3 Formally real fields......Page 318
8.4 The Witt ring of a field......Page 331
8.5 The symplectic group......Page 339
8.6 The orthogonal group......Page 346
8.7 Quadratic forms in characteristic 2......Page 353
Further exercises......Page 356
9.1 Rings of fractions......Page 358
9.2 Principal ideal domains......Page 364
9.3 Skew polynomials and Laurent series......Page 368
9.4 Goldie's theorem......Page 376
9.5 PI-algebras......Page 384
9.6 Varieties of PI-algebras and Regev's theorem......Page 389
9.7 Generic matrix rings and central polynomials......Page 393
9.8 Generalized polynomial identities......Page 398
Further exercises......Page 402
10.1 The endomorphism ring of a vector space......Page 404
10.2 The density theorem revisited......Page 407
10.3 Primitive rings......Page 410
10.4 Semiprimitive rings and the Jacobson radical......Page 413
10.5 Algebras without a unit element......Page 418
10.6 Semiprime rings and nilradicals......Page 422
10.7 Prime PI-algebras......Page 428
10.8 Simple algebras with minimal right ideals......Page 432
10.9 Firs and semifirs......Page 434
Further exercises......Page 439
11.1 Generalities......Page 441
11.2 The Dieudonné determinant......Page 445
11.3 Free fields......Page 452
11.4 Valuations on skew fields......Page 457
11.5 Pseudo-linear extensions......Page 464
Further exercises......Page 468
Bibliography......Page 470
List of notations......Page 473
Index......Page 477