Lectures on Rings and Modules

Cover......Page 1
Lectures on Rings and Modules......Page 3
Copyright Page......Page 4
Preface......Page 5
Contents......Page 7
1.1 Rings and related algebraic systems......Page 9
1.2 Subrings, homomorphisms, ideals......Page 16
1.3 Modules, direct products, and direct sums......Page 22
1.4 Classical isomorphism theorems......Page 27
2.1 Prime ideals in commutative rings......Page 35
2.2 Prime ideals in special commutative rings......Page 40
2.3 The complete ring of quotients of a commutative ring......Page 44
2.4 Rings of quotients of commutative semiprime rings......Page 50
2.5 Prime ideal spaces......Page 55
3.1 Primitive rings......Page 59
3.2 Radicals......Page 63
3.3 Completely reducible modules......Page 67
3.4 Completely reducible rings......Page 70
3.5 Artinian and Noetherian rings......Page 75
3.6 On lifting idempotents......Page 80
3.7 Local and semiperfect rings......Page 83
4.1 Projective modules......Page 88
4.2 Injective modules......Page 95
4.3 The complete ring of quotients......Page 102
4.4 Rings of endomorphisms of injective modules......Page 109
4.5 Regular rings of quotients......Page 114
4.6 Classical rings of quotients......Page 116
4.7 The Faith-Utumi Theorem......Page 122
5.1 Tensor products of modules......Page 125
5.2 Horn and 0 as functors......Page 129
5.3 Exact sequences......Page 134
5.4 Flat modules......Page 140
5.5 Torsion and extension products......Page 144
Appendixes......Page 153
Comments......Page 174
Bibliography......Page 180
Index......Page 189