Fundamental Concepts of Algebra (eBook)

Front Cover 1
Fundamental Concepts of Algebra 4
Copyright Page 5
Contents 8
Preface 6
Prerequisite knowledge and terminological conventions 10
Chapter I. Monoids 12
1. Definition of a monoid 12
2. Submonoids. Generators 17
3. Homomorphisms 19
4. Quotient monoids 22
5. Products 24
6. Free monoids 27
Exercises 31
Chapter II. Groups 34
1. Definition of a group 34
2. Subgroups 36
3. Homomorphisms. Quotient groups 38
4. Groups operating on a set 44
5.Products of groups 48
6. Free groups 49
Exercises 52
Chapter III. Rings and modules 58
1. Rings 58
2. Field of quotients 61
3. Modules 63
4. Submodules 65
5. Linear mappings 72
6. Products 78
7. Uniqueness theorems for semi-simple modules 80
8. Tensor products of modules 83
9. Free modules. Bases 89
10. Multilinear mappings 92
11. Transfer of basic rings 106
12. Vector spaces 111
13. Vector spaces in duality 115
14. The rank of a linear mapping 120
15. Matrices 121
16. Systems of linear equations 132
17. Graded modules 133
Exercises 137
Chapter IV. Algebras 146
1. Definition 146
2. Subalgebras 147
3. Homomorphisms 148
4. Products 149
5. Free algebra 150
Exercises 152
Chapter V. Associative algebras 154
1. Definitions 154
2. Graded algebras 158
3. Tensor algebras 160
4. Tensor products of graded algebras 163
5. Anticommutative algebras 167
6. Derivations 171
7. Exterior algebras 174
8. Grassmann algebras 179
9. The determinant of a matrix 185
10. Some applications of determinants 191
11. Existence of certain derivations 196
12. The trace of a matrix 201
13. Alternating multilinear mappings 202
14. The Pfaffian of an alternating bilinear form 203
15. Exterior algebras on vector spaces 209
16. Transfer of the basic ring 213
17. Commutative tensor products 220
18. Symmetric algebras 222
19. Polynomial algebras 230
Exercises 237
Index 247