An Introduction to Algebraic Structures

Superb, self-contained text covers sets and numbers, the elements of set theory, the real numbers, the theory of groups, group isomorphism and homomorphism, the theory of rings, and polynomial rings. Numerous examples and exercises. Ideal for one-semester course. 1969 edition.

A Professor Emeritus at the University of Illinois, Joseph Landin served as Head of the Department of Mathematics for 10 years.

1. Sets and Numbers
  I. THE ELEMENTS OF SET THEORY
    1. The Concept of Set
    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 Partititions
  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. Principle 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 Intermediates
  9. Substitution Homomorphisms
  10. Roots of Polynomials