Book:Nathan Jacobson/Lectures in Abstract Algebra/Volume I

Subject Matter

 * Abstract Algebra
 * Group Theory
 * Rings
 * Fields
 * Euclidean Domains
 * Polynomial Theory
 * Modules
 * Ideal Theory
 * Lattice Theory

Contents

 * Preface

Introduction: Concepts from Set Theory: The System of Natural Numbers

 * 1. Operations on sets
 * 2. Product sets, mappings
 * 3. Equivalence relations
 * 4. The natural numbers
 * 5. The system of integers
 * 6. The division process in $$I$$

Chapter I: Semi-Groups and Groups

 * 1. Definition and examples of semi-groups
 * 2. Non-associative binary compositions
 * 3. Generalized associative law. Powers
 * 4. Commutativity
 * 5. Identities and inverses
 * 6. Definition and examples of groups
 * 7. Subgroups
 * 8. Isomorphism
 * 9. Transformation groups
 * 10. Realization of a group as a transformation group
 * 11. Cyclic groups. Order of an element
 * 12. Elementary properties of permutations
 * 13. Coset decompositions of groups
 * 14. Invariant subgroups and factor groups
 * 15. Homomorphism of groups
 * 16. The fundamental theorem of homomorphism for groups
 * 17. Endomorphisms, automorphisms, center of a group
 * 18. Conjugate classes

Chapter II: Rings, Integral Domains and Fields

 * 1. Definition and examples
 * 2. Types of rings
 * 3. Quasi-regularity. The circle composition
 * 4. Matrix rings
 * 5. Quaternions
 * 6. Subrings generated by a set of elements. Center
 * 7. Ideals, difference rings
 * 8. Ideals and difference rings for the ring of integers
 * 9. Homomorphism of rings
 * 10. Anti-isomorphism
 * 11. Structure of the additive group of a ring. The characteristic of a ring
 * 12. Algebra of subgroups of the additive group of a ring. One-sided ideals
 * 13. The ring of endomorphisms of a commutative group
 * 14. The multiplications of a ring

Chapter III: Extensions of a Ring and Fields

 * 1. Imbedding of a ring in a ring with an identity
 * 2. Fields of fractions on a commutative integral domain
 * 3. Uniqueness of the field of fractions
 * 4. Polynomial rings
 * 5. Structure of polynomial rings
 * 6. Properties of the ring $$\mathfrak{A} \left[{x}\right]$$
 * 7. Simple extensions of a field
 * 8. Structure of any field
 * 9. The number of roots of a polynomial in a field
 * 10. Polynomials in several elements
 * 11. Symmetric polynomials
 * 12. Rings of functions

Chapter IV: Elementary Factorization Theory

 * 1. Factors, associates, irreducible elements
 * 2. Gaussian semi-groups
 * 3. Greatest common divisors
 * 4. Principal ideal domains
 * 5. Euclidean domains
 * 6. Polynomial extensions of Gaussian domains

Chapter V: Groups with Operators

 * 1. Definition and examples of groups with operators
 * 2. M-subgroups, M-factor groups and M-homomorphisms
 * 3. The fundamental theorem of homomorphisms for M-groups
 * 4. The correspondence between M-subgroups determined by a homomorphism
 * 5. The isomorphism theorems for M-groups
 * 6. Schreier's theorem
 * 7. Simple groups and the Jordan-Hölder theorem
 * 8. The chain conditions
 * 9. Direct products
 * 10. Direct products of subgroups
 * 11. Projections
 * 12. Decomposition into indecomposable groups
 * 13. The Krull-Schmidt theorem
 * 14. Infinite direct products

Chapter VI: Modules and Ideals

 * 1. Definitions
 * 2. Fundamental concepts
 * 3. Generators. Unitary modules
 * 4. The chain conditions
 * 5. The Hilbert basis theorem
 * 6. Noetherian rings. Prime and primary ideals
 * 7. Representation of an ideal as intersection of primary ideals
 * 8. Uniqueness theorems
 * 9. Integral dependence
 * 10. Integers of quadratic fields

Chapter VII: Lattices

 * 1. Partially ordered sets
 * 2. Lattices
 * 3. Modular lattices
 * 4. Schreier's theorem. The chain conditions
 * 5. Decomposition theory for lattices with ascending chain condition
 * 6. Independence
 * 7. Complemented modular lattices
 * 8. Boolean algebras


 * Index