Lectures in Abstract Algebra I : Basic Concepts / by N. Jacobson.

Format:

Book

Author/Creator:

Jacobson, Nathan, 1910-1999, author.

Contributor:

SpringerLink (Online service)

Series:

Graduate texts in mathematics 2197-5612 ; 30.

Graduate Texts in Mathematics, 2197-5612 ; 30

Language:

English

Subjects (All):

Algebra.

Local Subjects:

Algebra.

Physical Description:

1 online resource (217 pages).

Edition:

First edition 1951.

Contained In:

Springer Nature eBook

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 1951.

System Details:

text file PDF

Summary:

The present volume is the first of three that will be published under the general title Lectures in Abstract Algebra. These vol- umes are based on lectures which the author has given during the past ten years at the University of North Carolina, at The Johns Hopkins University, and at Yale "University. The general plan of the work IS as follows: The present first volume gives an introduction to abstract algebra and gives an account of most of the important algebraIc concepts. In a treatment of this type it is impossible to give a comprehensive account of the topics which are introduced. Nevertheless we have tried to go beyond the foundations and elementary properties of the algebraic sys- tems. This has necessitated a certain amount of selection and omission. We feel that even at the present stage a deeper under- standing of a few topics is to be preferred to a superficial under- standing of many. The second and third volumes of this work will be more special- ized in nature and will attempt to give comprehensive accounts of the topics which they treat. Volume II will bear the title Linear Algebra and will deal with the theorv of vectQ!_JlP. -a. ces. . . . . Volume III, The Theory of Fields and Galois Theory, will be con- cerned with the algebraic structure offieras and with valuations of fields. All three volumes have been planned as texts for courses.

Contents:

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

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 a group

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

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 charateristic 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

III: Extensions of Rings and Fields

1. Imbedding of a ring in a ring with an identity

2. Field of fractions of a commutative integral domain

3. Uniqueness of the field of fractions

4. Polynomial rings

5. Structure of polynomial rings

6. Properties of the ring U[x]

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

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

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 homomorphism 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

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

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.

Other Format:

Printed edition:

ISBN:

9781468473018

Access Restriction:

Restricted for use by site license.