An introduction to algebraic structures

Description

Title Preface Contents 1. Sets and Numbers I. The Elements of Set Theory 1. The Concept ofSet 2. Constants, Variables and Related Matters 3. Subsets and Equality of Sets 4. The Algebra of Sets; The Empty Set 5. A Notation for Sets 6. Generalized Intersection and Union 7. Ordered Pairs and Cartesian Products 8. Functions (or Mappings) 9. A Classification of Mappings 10. Composition of Mappings 11. Equivalence Relations and Partitions II. The Real Numbers 12. Introduction 13. The Real Numbers 14. The Natural Numbers 15. The Integers 16. The Rational Numbers 17. The Complex Numbers 2. The Theory of Groups 1. The Group Concept 2. Some Simple Consequences of the Definition of Group 3. Powers of Elements in a Group 4. Order of a Group; Order of a Group Element 5. Cyclic Groups 6. The Symmetric Groups 7. Cycles; Decomposition of Permutations Into Disjoint Cycles 8. Full Transformation Groups 9. Restrictions of Binary Operations 10. Subgroups 11. A Discussion of Subgroups 12. The Alternating Group 13. The Congruence of Integers 14. The Modular Arithmetics 15. Equivalence Relations and Subgroups 16. Index of a Subgroup 17. Stable Relations, Normal Subgroups, Quotient Groups 18. Conclusion 3. Group Isomorphism and Homomorphism 1. Introduction 2. Group Isomorphism; Examples, Definitions, and Simplest Properties 3. The Isomorphism Theorem for the Symmetric Groups 4. The Theorem of Cayley 5. Group Homomorphisms 6. A Relation Between Epimorphisms and Isomorphisms 7. Endomorphisms of a Group 4. The Theory of Rings 1. Introduction 2. Definition of Ring 3. Some Properties of Rings 4. The Modular Arithmetics, Again 5. Integral Domains 6. Fields 7. Subrings 8. Ring Homomorphisms 9. Ideals 10. Residue Class Rings 11. Some Basic Homomorphism Theorems 12. Principal Ideal and Unique Factorization Domains 13. Prime and Maximal Ideals 14. The Quotient Field of an Integral Domain 5. Polynomial Rings 1. Introduction; The Concept of Polynomial Ring 2. Indeterminates 3. Existence of Indeterminates 4. Polynomial Domains Over a Field 5. Unique Factorization in Polynomial Domains 6. Polynomial Rings in Two Indeterminates 7. Polynomial Functions and Polynomials 8. Some Characterizations of Indeterminates 9. Substitution Homomorphisms 10. Roots of Polynomials Index Back Cover