Algebra

1 Set theory.- 1.1 Sets.- 1.2 Operations on sets.- 1.3 Relations.- 1.4 Quotient sets.- 1.5 Functions.- 1.6 Composition of functions.- 1.7 A factorization of a function.- 1.8 The symmetric group.- 2 Rings: Basic theory.- 2.1 Binary operations.- 2.2 The ring.- 2.3 Special rings.- 2.4 Subrings.- 2.5 Morphisms.- 2.6 Quotient rings.- 2.7 Morphisms and quotient rings.- 2.8 Ideals.- 3 Rings: Natural numbers and integers.- 3.1 The Peano axioms.- 3.2 Addition of natural numbers.- 3.3 Multiplication of natural numbers.- 3.4 Further properties of.- 3.5 Construction of the integers.- 3.6 Embedding ? in the integers.- 3.7 Ordered integral domains.- 3.8 A characterization of the integers.- 4 Rings: Applications of the integers.- 4.1 Finite sets.- 4.2 Generalized associative, commutative, and distributive theorems.- 4.3 The division algorithm for the integers.- 4.4 Multiples and exponents in a ring.- 4.5 The field of fractions.- 4.6 Characteristic of a ring.- 5 Rings: Polynomials and factorization.- 5.1 The ring of polynomials.- 5.2 A formal definition of a polynomial ring.- 5.3 Polynomial functions.- 5.4 Euclidean and principal ideal domains.- 5.5 Factorization in principal ideal domains.- 5.6 Greatest common divisor.- 5.7 Unique factorization domains.- 5.8 Field extensions and complex numbers.- 6 Linear algebra: Modules.- 6.1 Function spaces, modules, and vector spaces.- 6.2 Submodules.- Appendix 6A A method for solution of linear equations.- 6.3 Quotient modules.- 6.4 Morphisms.- 6.5 Products and direct sums.- 6.6 Families and matrices.- 6.7 Bases.- 6.8 The coordinate morphism.- 6.9 Morphisms and bases, kernel, and range.- 6.10 Vector spaces.- Appendix 6B The existence of a basis for a vector space.- Appendix 6C Equicardinality of infinite bases of a vector space.- Appendix 6DDimension of a module over a commutative unitary ring.- 7 Linear algebra: The module of morphisms.- 7.1 ?(M, M?), the module of morphisms.- 7.2 Composition of morphisms, the endomorphism algebra ?(M).- 7.3 Matrix calculation of morphisms.- 7.4 Change of basis.- 7.5 The dual space.- 7.6 Linear equations.- 7.7 Determinants.- 8 Abstract systems.- 8.1 Algebraic systems.- 8.2 Algebraic subsystems.- 8.3 Morphisms.- 8.4 Congruences and quotient systems.- 8.5 Products and sums.- 9 Monoids and groups.- 9.1 Monoids, unitary monoids, cancellative monoids, and groups.- 9.2 Congruences and quotient systems.- 9.3 Morphisms.- 9.4 Cyclic groups and order.- 9.5 Products.- 10 Linear algebra: Modules over principal domains and similarity.- 10.1 Cyclic modules.- 10.2 Invariant factors.- 10.3 Linear equations in a principal domain.- 10.4 A direct sum resolution of a finitely generated module.- 10.5 Similarity and canonical forms.- 10.6 The characteristic polynomial and characteristic values.- Selected references.- Answers to questions.- Index of symbols.