Groups, Rings, Modules

PREFACE
PART ONE
Chapter 1 SETS AND MAPS
INTRODUCTION
1. SETS AND SUBSETS
2. MAPS
3. ISOMORPHISMS OF SETS
4. EPIMORPHISMS AND MONOMORPHISMS
5. THE IMAGE ANALYSIS OF A MAP
6. THE COIMAGE ANALYSIS OF A MAP
7. DESCRIPTION OF SURJECTIVE MAPS
8. EQUIVALENCE RELATIONS
9. CARDINALITY OF SETS
10. ORDERED SETS
11. AXIOM OF CHOICE
12. PRODUCTS AND SUMS OF SETS
EXERCISES
Chapter 2 MONOIDS AND GROUPS
1. MONOIDS
2. MORPH ISMS OF MONOIDS
3. SPECIAL TYPES OF MORPHISMS
4. ANALYSES OF MORPHISMS
5. DESCRIPTION OF SURJECTIVE MORPHISMS
6. GROUPS AND MORPHISMS OF GROUPS
7. KERNELS OF MORPHISMS OF GROUPS
8. GROUPS OF FRACTIONS
9. THE INTEGERS
10. FINITE AND INFINITE SETS
EXERCISES
Chapter 3 CATEGORIES
1. CATEGORIES
2. MORPHISMS
3. PRODUCTS AND SUMS
EXERCISES
Chapter 4 RINGS
1. CATEGORY OF RINGS
2. POLYNOMIAL RINGS
3. ANALYSES OF RING MORPHISMS
4. IDEALS
5. PRODUCTS OF RINGS
EXERCISES
PART TWO
Chapter 5 UNIQUE FACTORIZATION DOMAINS
1. DIVISIBILITY
2. INTEGRAL DOMAINS
3. UNIQUE FACTORIZATION DOMAINS
4. DIVISIBILITY IN UFD'S
5. PRINCIPAL IDEAL DOMAINS
6. FACTOR RINGS OF PID'S
7. DIVISORS
8. LOCALIZATION IN INTEGRAL DOMAINS
9. A CRITERION FOR UNIQUE FACTORIZATION
10. WHEN R[X]ISAUFD
EXERCISES
Chapter 6 GENERAL MODULE THEORY
1. CATEGORY OF MODULES OVER A RING
2. THE COMPOSITION MAPS IN Mod(R)
3. ANALYSES OF R-MODULE MORPHISMS
4. EXACT SEQUENCES
5. ISOMORPHISM THEOREMS
6. NOETHERIAN AND ARTINIAN MODULES
7. FREE R-MODULES
8. CHARACTERIZATION OF DIVISION RINGS
9. RANK OF FREE MODULES
10. COMPLEMENTARY SUBMODULES OF A MODULE
11. SUMS OF MODULES
12. CHANGE OF RINGS
13. TORSION MODULES OVER PID'S
14. PRODUCTS OF MODULES
EXERCISES
Chapter 7 SEMISIMPLE RINGS AND MODULES
1. SIMPLE RINGS
2. SEMISIMPLE MODULES
3. PROJECTIVE MODULES
4. THE OPPOSITE RING
EXERCISES
Chapter 8 ARTIN IAN RINGS
1. IDEMPOTENTS IN LEFT ARTINIAN RINGS
2. THE RADICAL OF A LEFT ARTINIAN RING
3. THE RADICAL OF AN ARBITRARY RING
EXERCISES
PART THREE
Chapter 9 LOCALIZATION AND TENSOR PRODUCTS
1. LOCALIZATION OF RINGS
2. LOCALIZATION OF MODULES
3. APPLICATIONS OF LOCALIZATION
4. TENSOR PRODUCTS
5. MORPHISMS OF TENSOR PRODUCTS
6. LOCALLY FREE MODULES
EXERCISES
Chapter 10 PRINCIPAL IDEAL DOMAINS
1. SUBMODULES OF FREE MODULES
2. FREE SUBMODULES OF FREE MODULES
3. FINITELY GENERATED MODULES OVER PID'S
4. INJECTIVE MODULES
5. THE FUNDAMENTAL THEOREM FOR PID'S
EXERCISES
Chapter 11 APPLICATIONS OF THE FUNDAMENTAL THEOREM
1. DIAGONALIZATION
2. DETERMINANTS
3. MATRICES
4. FURTHER APPLICATIONS OF THE FUNDAMENTAL THEOREM
5. CANONICAL FORMS
EXERCISES
PART FOUR
Chapter 12 ALGEBRAIC FIELD EXTENSIONS
1. ROOTS OF POLYNOMIALS
2. ALGEBRAIC ELEMENTS
3. MORPHISMS OF FIELDS
4. SEPARABILITY
5. GALOIS EXTENSIONS
EXERCISES
Chapter 13 DEDEKIND DOMAINS
1. DEDEKIND DOMAINS
2. INTEGRAL EXTENSIONS
3. CHARACTERIZATIONS OF DEDEKIND DOMAINS
4. IDEALS
5. FINITELY GENERATED MODULES OVER DEDEKIND DOMAINS
EXERCISES
INDEX