## Lectures in Abstract Algebra I: Basic ConceptsThe 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

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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### Common terms and phrases

additive group arbitrary ascending chain condition associated primes automorphism binary composition called cancellation law clear coefficients commutative group commutative ring complex numbers composition series consider contains correspondence coset cyclic group decomposition define definition deg f(x denote determined direct product distributive laws division ring divisor element of 91 endomorphism example EXERCISES exists factor group field finite number following Theorem function Gaussian group of order groups with operators Hence holds implies indecomposable integral domain intersection invariant M-subgroups inverse irreducible elements irredundant isomorphism kernel left ideal Lemma M-group matrix modular lattice module morphism natural numbers Noetherian ring non-zero elements obtain partially ordered set permutation polynomial positive integers primary ideals principal ideal domain Proof prove the following rational numbers real numbers relative result ring 91 satisfies semi-group Show subfield submodule subring subset suppose symmetric theory tion transcendental transformation group uniqueness vector verify zero-divisor