Book:Nathan Jacobson/Lectures in Abstract Algebra: I. Basic Concepts

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Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts

Published $1951$, Van Nostrand.

Subject Matter



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


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