Lectures in Abstract Algebra I: Basic Concepts

ISBN-13: 9781468473032 / Angielski / Miękka / 2012 / 217 str.

N. Jacobson

Opis

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.

Spis treści

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.