Kurosh / Sneddon / Stark Lectures in General Algebra

1. Auflage 2014
ISBN: 978-1-4831-4957-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 374 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

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.

Autoren/Hrsg.

Kurosh, A. G.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Lectures in General Algebra;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE;8
6;CHAPTER ONE. RELATIONS;12
6.1;§ 1. Sets;12
6.2;§ 2. Binary relations;14
6.3;§ 3. Equivalence relations;18
6.4;§ 4. Partial ordering;20
6.5;§ 5. The minimum condition;22
6.6;§ 6. Theorems equivalent to the axiom of choice;26
7;CHAPTER TWO. GROUPS AND RINGS;31
7.1;§ 1. Groupoids, semigroups, groups;31
7.2;§ 2. Rings, skew fields, fields;36
7.3;§ 3. Subgroups, subring;45
7.4;§ 4. Isomorphism;49
7.5;§ 5. Embedding semigroups in groups and rings in skew fields;55
7.6;§ 6. Non-associative skew fields, quasi-groups. Isotopy;63
7.7;§ 7. Normal subgroups, ideals;69
7.8;§ 8. Gaussian semigroups;77
7.9;§ 9. Gaussian rings;84
7.10;§ 10. Dedekind rings;92
8;CHAPTER THREE. UNIVERSAL ALGEBRAS GROUPS WITH MULTI-OPERATORS;102
8.1;§ 1. Universal algebras. Homomorphisms;102
8.2;§ 2. Groups with multi-operators;108
8.3;§ 3. Automorphisms, endomorphisms.The field of p-adic numbers;119
8.4;§ 4. Normal and composition series;130
8.5;§ 5. Abelian, nilpotent and soluble O-groups;136
8.6;§ 6. Primitive classes of universal algebras;144
8.7;§ 7. Free universal algebras;148
8.8;§ 8. Free products of groups;158
9;CHAPTER FOUR. LATTICES;172
9.1;§ 1. Lattices, complete lattices;172
9.2;§ 2. Modular lattices;181
9.3;§ 3. Direct unions. The Schmidt-Ore theorem;188
9.4;§ 4. Direct decompositions of O-groups;197
9.5;§ 5. Complete direct sums of universal algebras;202
9.6;§ 6. Distributive lattices;207
10;CHAPTER FIVE. OPERATOR GROUPS AND RINGS. MODULES. LINEAR ALGEBRAS;212
10.1;§ 1. Operator groups and rings;212
10.2;§ 2. Free modules. Abelian groups;220
10.3;§ 3. Vector spaces over skew fields;227
10.4;§ 4. Rings of linear transformations;232
10.5;§ 5. Simple rings. Jacobson's theorem;238
10.6;§ 6. Linear algebras. The algebra of quaternions and the Cayley algebra;246
10.7;§ 7. Alternative rings. Artin's theorem;254
10.8;§ 8. A generalization of Frobenius' theorem;260
10.9;§ 9. The Birkhoff-Witt theorem on Lie algebras;269
10.10;§ 10. Derivations. Differential rings;275
11;CHAPTER SIX. ORDERED AND TOPOLOGICAL GROUPS AND RINGS. NORMED RINGS;283
11.1;§ 1. Ordered groups;283
11.2;§ 2. Ordered rings;289
11.3;§ 3. Archimedean groups and rings;295
11.4;§ 4. Normed rings;303
11.5;§ 5. Valuated fields;309
11.6;§ 6. Albert's theorem on normed algebras;315
11.7;§ 7. Closure. Topological spaces;321
11.8;§ 8. Special types of topological spaces;329
11.9;§ 9. Topological groups;332
11.10;§ 10. The connection between topologies and norms in rings and skew fields;339
11.11;§ 11. Galois correspondences. The fundamental theorem of Galois theory;348
12;BIBLIOGRAPHY;358
13;INDEX;364
14;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;374