Lectures in Abstract Algebra I: Basic Concepts

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 217 pages
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.
 

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Contents

CONCEPTS FROM SET THEORY
1
SEMIGROUPS AND GROUPS
15
Commutativity
21
Transformation groups
27
Elementary properties of permutations
34
Invariant subgroups and factor groups
40
Conjugate classes
47
RINGs INTEGRAL DOMAINS AND FIELDs SECTION PAGE 1 Definition and examples
49
Greatest common divisors
118
Principal ideal domains
121
SECTION PAGE 5 Euclidean domains
122
Polynomial extensions of Gaussian domains
124
GROUPS WITH OPERATORS 1 Definition and examples of groups with operators
128
Msubgroups Mfactor groups and Mhomomorphisms
130
The fundamental theorem of homomorphism for Mgroups
132
The correspondence between Msubgroups determined by a homomorphism
133

Types of rings
53
Quasiregularity The circle composition
55
Matrix rings
56
Quaternions
60
Subrings generated by a set of elements Center
63
Ideals difference rings
64
Ideals and difference rings for the ring of integers
66
Homomorphism of rings
68
Antiisomorphism
71
Structure of the additive group of a ring The charateristic of a ring
74
Algebra of subgroups of the additive group of a ring One sided ideals
75
The ring of endomorphisms of a commutative group
78
The multiplications of a ring
82
EXTENSIONS OF RINGS AND FIELDS 1 Imbedding of a ring in a ring with an identity
84
Field of fractions of a commutative integral domain
87
Uniqueness of the field of fractions
91
Polynomial rings
92
Structure of polynomial rings
96
Properties of the ring 21x
97
Simple extensions of a field
100
Structure of any field
103
The number of roots of a polynomial in a field
104
Polynomials in several elements
105
Symmetric polynomials
107
Rings of functions
110
ELEMENTARY FACTORIZATION THEORY 1 Factors associates irreducible elements
114
Gaussian semigroups
115
The isomorphism theorems for Mgroups
135
Schreiers theorem
137
Simple groups and the JordanHölder theorem
139
The chain conditions
142
Direct products
144
Direct products of subgroups
145
Projections
149
Decomposition into indecomposable groups
152
The KrullSchmidt theorem
154
Infinite direct products
159
MODULES AND IDEALS 1 Definitions
162
Fundamental concepts
164
Generators Unitary modules
166
The chain conditions
168
The Hilbert basis theorem
170
Noetherian rings Prime and primary ideals
172
Representation of an ideal as intersection of primary ideals
175
Uniqueness theorems
177
Integral dependence
181
Integers of quadratic fields
184
LATTICES 1 Partially ordered sets
187
Lattices
189
Modular lattices
193
Schreiers theorem The chain conditions
197
SECTION PAGE
201
Boolean algebras
207
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