An introduction to algebraic structures

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