Contents
Preface ...................................................................................................................... ix
Conventions on Terminology ................................................................................ xi
1. Sets
1.1 Finite, Countable and Uncountable Sets ........................................... . 1
1.2 Zorn's Lemma and Well-ordered Sets ............................................... . 8
1.3 Graphs ................................................................................................... . 15
2. Groups
2.1 Definition and Basic Properties........................................................... 25
2.2 Permutation Groups ............................................................................. 32
2.3 The Isomorphism Theorems......................... ....................................... 34
2.4 Soluble and Nilpotent Groups............................................................. 37
2.5 Commutators......................................................................................... 42
2.6 The Frattini Subgroup and the Fitting Subgroup ............................. 46
3. Lattices and Categories
3.1 Definitions; Modular and Distributive Lattices ................................. 51
3.2 Chain Conditions .................................................................................. 60
3.3 Categories............................................................................................... 65
3.4 Boolean Algebras................................................................................... 70
4. Rings and Modules
4.1 The Definitions Recalled .......................... ............................................ 79
4.2 The Category of Modules over a Ring ............................................... 84
4.3 Semisimple Modules............................................................................. 91
4.4 Matrix Rings ........................ ................................ ................. ................. 96
4.5 Direct Products of Rings...................................................................... 101
4.6 Free Modules ......................................................................................... 105
4.7 Projective and Injective Modules ...................... .................................. 110
4.8 The Tensor Product of Modules ......................................................... 117
4.9 Duality of Finite Abelian Groups........................................................ 125
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Preface ...................................................................................................................... ix
Conventions on Terminology ................................................................................ xi
1. Sets
1.1 Finite, Countable and Uncountable Sets ........................................... . 1
1.2 Zorn's Lemma and Well-ordered Sets ............................................... . 8
1.3 Graphs ................................................................................................... . 15
2. Groups
2.1 Definition and Basic Properties........................................................... 25
2.2 Permutation Groups ............................................................................. 32
2.3 The Isomorphism Theorems......................... ....................................... 34
2.4 Soluble and Nilpotent Groups............................................................. 37
2.5 Commutators......................................................................................... 42
2.6 The Frattini Subgroup and the Fitting Subgroup ............................. 46
3. Lattices and Categories
3.1 Definitions; Modular and Distributive Lattices ................................. 51
3.2 Chain Conditions .................................................................................. 60
3.3 Categories............................................................................................... 65
3.4 Boolean Algebras................................................................................... 70
4. Rings and Modules
4.1 The Definitions Recalled .......................... ............................................ 79
4.2 The Category of Modules over a Ring ............................................... 84
4.3 Semisimple Modules............................................................................. 91
4.4 Matrix Rings ........................ ................................ ................. ................. 96
4.5 Direct Products of Rings...................................................................... 101
4.6 Free Modules ......................................................................................... 105
4.7 Projective and Injective Modules ...................... .................................. 110
4.8 The Tensor Product of Modules ......................................................... 117
4.9 Duality of Finite Abelian Groups........................................................ 125
v
