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

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## Nathan Jacobson:

## Nathan Jacobson: *Lectures in Abstract Algebra, Volume $\text { I }$: Basic Concepts*

Published $\text {1951}$, **Van Nostrand**

### Subject Matter

- Abstract Algebra
- Group Theory
- Ring Theory
- Field Theory
- Euclidean Domains
- Polynomial Theory
- Module Theory
- 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 $\text I$: 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 $\text {II}$: 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 $\text {III}$: 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 $\text {IV}$: 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 $\text V$: 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-HÃ¶lder 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 $\text {VI}$: 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 $\text {VII}$: 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