# Lectures in General Algebra

## 1st Edition

**Editors:**I. N. Sneddon M. Stark S. Ulam

**Authors:**A. G. Kurosh

**eBook ISBN:**9781483149578

**Imprint:**Pergamon

**Published Date:**1st January 1965

**Page Count:**374

**View all volumes in this series:**International Series in Pure and Applied Mathematics

## Table of Contents

Preface

Chapter One. Relations

§ 1 Sets

§ 2 Binary Relations

§ 3 Equivalence Relations

§ 4 Partial Ordering

§ 5 The Minimum Condition

§ 6 Theorems Equivalent to the Axiom of Choice

Chapter Two. Groups and Rings

§ 1 Groupoids, Semigroups, Groups

§ 2 Rings, Skew Fields, Fields

§ 3 Subgroups, Subrings

§ 4 Isomorphism

§ 5 Embedding of Semigroups in Groups and Rings in Skew Fields

§ 6 Non-Associative Skew Fields, Quasi Groups. Isotopy

§ 7 Normal Subgroups, Ideals

§ 8 Gaussian Semigroups

§ 9 Gaussian Rings

§ 10 Dedekind Rings

Chapter Three. Universal Algebras. Groups with Multi-Operators

§ 1 Universal Algebras. Homomorphisms

§ 2 Groups with Multi-Operators

§ 3 Automorphisms, Endomorphisms. The Field of P-Adic Numbers

§ 4 Normal and Composition Series

§ 5 Abelian, Nilpotent and Soluble H-Groups

§ 6 Primitive Classes of Universal Algebras

§ 7 Free Universal Algebras

§ 8 Free Products of Groups

Chapter Four. Lattices

§ 1 Lattices, Complete Lattices

§ 2 Modular Lattices

§ 3 Direct Unions. The Schmidt-Ore Theorem

§ 4 Direct Decompositions of Ω-Groups

§ 5 Complete Direct Sums of Universal Algebras

§ 6 Distributive Lattices

Chapter Five. Operator Groups and Rings. Modules. Linear Algebras

§ 1 Operator Groups and Rings

§ 2 Free Modules. Abelian Groups

§ 3 Vector Spaces Over Skew Fields

§ 4 Rings of Linear Transformations

§ 5 Simple Rings. Jacobson's Theorem

§ 6 Linear Algebras. The Algebra of Quaternions and the Cayley Algebra

§ 7 Alternative Rings. Artin's Theorem

§ 8 A Generalization of Frobenius' Theorem

§ 9 The Birkhoff-Witt Theorem on Lie Algebras

§ 10 Derivations. Differential Rings

Chapter Six. Ordered and Topological Groups and Rings. Normed Rings

§ 1 Ordered Groups

§ 2 Ordered Rings

§ 3 Archimedean Groups and Rings

§ 4 Normed Rings

§ 5 Valuated Fields

§ 6 Albert's Theorem on Normed Algebras

§ 7 Closure. Topological Spaces

§ 8 Special Types of Topological Spaces

§ 9 Topological Groups

§ 10 The Connection between Topologies and Norms in Rings and Skew Fields

§ 11 Galois Correspondences. The Fundamental Theorem of Galois Theory

Bibliography

Index

Other Titles in the Series in Pure and Applied Mathematics

## Description

Lectures in General Algebra is a translation from the Russian and is based on lectures on specialized courses in general algebra at Moscow University.

The book starts with the basics of algebra. The text briefly describes the theory of sets, binary relations, equivalence relations, partial ordering, minimum condition, and theorems equivalent to the axiom of choice. The text gives the definition of binary algebraic operation and the concepts of groups, groupoids, and semigroups. The book examines the parallelism between the theory of groups and the theory of rings; such examinations show the convenience of constructing a single theory from the results of group experiments and ring experiments which are known to follow simple corollaries. The text also presents algebraic structures that are not of binary nature. From this parallelism arise other concepts, such as that of the lattices, complete lattices, and modular lattices. The book then proves the Schmidt-Ore theorem, and also describes linear algebra, as well as the Birkhoff-Witt theorem on Lie algebras. The text also addresses ordered groups, the Archimedean groups and rings, and Albert's theorem on normed algebras.

This book can prove useful for algebra students and for professors of algebra and advanced mathematicians.

## Details

- No. of pages:
- 374

- Language:
- English

- Copyright:
- © Pergamon 1965

- Published:
- 1st January 1965

- Imprint:
- Pergamon

- eBook ISBN:
- 9781483149578