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

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-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 $\text {VI}$: 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

Next

Cited by

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra