# Book:Nathan Jacobson/Lectures in Abstract Algebra

## Nathan Jacobson: Lectures in Abstract Algebra

Published $\text {1951 - 1964}$.

## Subject Matter

### Volume $\text I$: Basic Concepts

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

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

### Volume $\text {II}$: Linear Algebra

Preface

Chapter $\text I$: Finite Dimensional Vector Spaces
1. Abstract vector spaces
2. Right vector spaces
3. $\mathfrak o$-modules
4. Linear dependence
5. Invariance of dimensionality
6. Bases and matrices
7. Applications to matrix theory
8. Rank of a set of vectors
9. Factor spaces
10. Algebra of subspaces
11. Independent subspaces, direct sums

Chapter $\text {II}$: Linear Transformations
1. Definition and examples
2. Composition of linear transformations
3. The matrix of a linear transformation
4. Composition of matrices
5. Change of basis. Equivalence and similarity of matrices
6. Rank space and null space of a linear transformation
7. Systems of linear equations
8. Linear transformations in right vector spaces
9. Linear functions
10. Duality between a finite dimensional linear space and its conjugate space
11. Transpose of a linear transformation
12. Matrices of the transpose
13. Projections

Chapter $\text {III}$: The Theory of a Single Linear Transformation
1. The minimum polynomial of a linear transformation
2. Cyclic subspaces
3. Existence of a vector whose order is the minimum polynomial
4. Cyclic linear transformations
5. The $\Phi \left[{\lambda}\right]$-module determined by a linear transformation
6. Finitely generated $\mathfrak o$-modules, $\mathfrak o$, a principal ideal domain
7. Normalization of the generators of $\mathfrak F$ and of $\mathfrak R$
8. Equivalence of matrices with elements in a principal ideal domain
9. Structure of finitely generated $\mathfrak o$-modules
10. Invariance theorems
11. Decomposition of a vector space relative to a linear transformation
12. The characteristic and minimum polynomials
13. Direct proof of Theorem 13
14. Formal properties of the trace and the characteristic polynomial
15. The ring of $\mathfrak o$-endomorphisms of a cyclic $\mathfrak o$-module
16. Determination of the ring of $\mathfrak o$-endomorphisms of a finitely generated $\mathfrak o$-module, $\mathfrak o$ principal
17. The linear transformations which commute with a given linear transformation
18. The center of the ring $\mathfrak B$

Chapter $\text {IV}$: Sets of Linear Transformations
1. Invariant subspaces
2. Induced linear transformations
3. Composition series
4. Decomposability
5. Complete reducibility
6. Relation to the theory of operator groups and the theory of modules
7. Reducibility, decomposability, complete reducibility for a single linear transformation
8. The primary components of a space relative to a linear transformation
9. Sets of commutative linear transformations

Chapter $\text V$: Bilinear Forms
1. Bilinear forms
2. Matrices of a bilinear form
3. Non-degenerate forms
4. Transpose of a linear transformation relative to a pair of bilinear forms
5. Another relation between linear transformations and bilinear forms
6. Scalar products
7. Hermitian scalar products
8. Matrices of hermitian scalar products
9. Symmetric and hermitian scalar products over special division rings
10. Alternate scalar products
11. Witt's theorem
12. Non-alternate skew-symmetric forms

Chapter $\text {VI}$: Euclidean and Unitary Spaces
1. Cartesian bases
2. Linear transformations and scalar products
3. Orthogonal complete reducibility
4. Symmetric, skew and orthogonal linear transformations
5. Canonical matrices for symmetric and skew linear transformations
6. Commutative symmetric and skew linear transformations
7. Normal and orthogonal linear transformations
8. Semi-definite transformations
9. Polar factorization of an arbitrary linear transformation
10. Unitary geometry
11. Analytic functions of linear transformations

Chapter $\text {VII}$: Products of Vector Spaces
1. Product groups of vector spaces
2. Direct products of linear transformations
3. Two-sided vector spaces
4. The Kronecker product
5. Kronecker products of linear transformations and of matrices
6. Tensor spaces
7. Symmetry classes of tensors
8. Extension of the field of a vector space
9. A theorem on similarity of sets of matrices
10. Alternative definition of an algebra. Kronecker product of algebras

Chapter $\text {VIII}$: The Ring of Linear Transformations
1. Simplicity of $\mathfrak L$
2. Operator methods
3. The left ideals of $\mathfrak L$
4. Right ideals
5. Isomorphisms of rings of linear transformations

Chapter $\text {IX}$: Infinite Dimensional Vector Spaces
1. Existence of a basis
2. Invariance of dimensionality
3. Subspaces
4. Linear transformations and matrices
5. Dimensionality of the conjugate space
6. Finite topology for linear transformations
7. Total subspaces of $\mathfrak R^*$
8. Dual subspaces. Kronecker products
9. Two-sided ideals in the ring of linear transformations
10. Dense rings of linear transformations
11. Isomorphism theorems
12. Anti-automorphisms and scalar products
13. Schur's lemma. A general density theorem
14. Irreducible algebras of linear transformations

Index

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