Author | Sigler, L. E. author |
---|---|
Title | Algebra [electronic resource] / by L. E. Sigler |
Imprint | New York, NY : Springer New York, 1976 |
Connect to | http://dx.doi.org/10.1007/978-1-4613-9410-5 |
Descript | 419 p. online resource |
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 6D Dimension 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