1．Generalities concerning modules 1
1．1 Left modules and right modules 1
1．2 Submodules 3
1．3 Factor modules 3
1．4 A-homomorphisms 3
1．5 Some different types of A-homomorphisms 4
1．6 Induced mappings 5
1．7 Images and kernels 6
1．8 Modules generated by subsets 7
1．9 Direct products and direct sums 9
1．10 Abbreviated notations 12
1．11 Sequences of A-homomorphisms 13
2．Tensor products and groups of homomorphisms 16
2．1 The definition of tensor products 16
2．2 Tensor products over commutative rings 17
2．3 Continuation of the general discussion 18
2．4 Tensor products of homomorphisms 19
2．5 The principal properties of HomA(B，C) 24
3．Categories and functors 30
3．1 Abstract mappings 30
3．2 Categories 31
3．3 Additive and A-categories 32
3．4 Equivalences 32
3．5 The categories ？LΛ and ？RΛ 33
3．6 Functors of a single variable 33
3．7 Functors of several variables 34
3．8 Natural transformations of functors 35
3．9 Functors of modules 36
3．10 Exact functors 38
3．11 Left exact and right exact functors 40
3．12 Properties of right exact functors 41
3．13 A⊕ΛA C and HomA(B，C)as functors 44
4．Homology functors 46
4．1 Diagrams over a ring 46
4．2 Translations of diagrams 47
4．3 Images and kernels as functors 48
4．4 Homology functors 52
4．5 The connecting homomorphism 54
4．6 Complexes 50
4．7 Homotopic translations 62
5．Projective and infective modules 63
5．1 Projective modules 63
5．2 Injective modules 67
5．3 An existence theorem for injective modules 71
5．4 Complexes over a module 75
5．5 Properties of resolutions of modules 77
5．6 Properties of resolutions of sequences 80
5．7 Further results on resolutions of sequences 84
6．Derived functors 90
6．1 Functors of complexes 90
6．2 Functors of two complexes 94
6．3 Right-derived functors 99
6．4 Left-derived functors 109
6．5 Connected sequences of functors 113
7．Torsion and extension functors 121
7．1 Torsion functors 121
7．2 Basic properties of torsion functors 123
7．3 Extension functors 128
7．4 Basic properties of extension functors 130
7．5 The homological dimension of a module 134
7．6 Global dimension 138
7．7 Noetherian rings 144
7．8 Commutative Noetherian rings 148
7．9 Global dimension of Noetherian rings 149
8．Some useful identities 155
8．1 Bimodules 155
8．2 General principles 156
8．3 The associative law for tensor products 160
8．4 Tensor products over commutative rings 161
8．5 Mixed identities 164
8．6 Rings and modules of fractions 167
9．Commutative Noetherian rings of finite global dimension 174
9．1 Some special cases 174
9．2 Reduction of the general problem 184
9．3 Modules over local rings 189
9．4 Some auxiliary results 202
9．5 Homological codimension 204
9．6 Modules of finite homological dimension 205
10．Homology and cohomology theories of groups and monoids 211
10．1 General remarks concerning monoids and groups 211
10．2 Modules with respect to monoids and groups 214
10．3 Monoid-rings and group-rings 215
10．4 The functors AG and AG 217
10．5 Axioms for the homology theory of monoids 219
10．6 Axioms for the cohomology theory of monoids 221
10．7 Standard resolutions of Z 223
10．8 The first homology group 229
10．9 The first cohomology group 230
10．10 The second cohomology group 238
10．11 Homology and cohomology in special cases 244
10．12 Finite groups 249
10．13 The norm of a homomorphism 252
10．14 Properties of the complete derived sequence 256
10．15 Complete free resolutions of Z 259
Notes 266
References 278
Index 281