Author(s): A. J. Berrick, M. E. Keating
Publisher: Cambridge
Year: 2000
Title page
PREFACE
1 BASICS
1.1 RINGS
1.1.1 The definition
1.1.2 Nonunital rings
1.1.3 Subrings
1.1.4 Ideals
1.1.5 Generators
1.1.6 Homomorphisms
1.1.8 Residue rings
1.1.10 The characteristic
1.1.11 Units
1.1.12 Constructing the field of fractions
1.1.13 Noncommutative polynomials
Exercises
1.2 MODULES
1.2.1 The definition
1.2.2 Some first examples
1.2.3 Bimodules
1.2.4 Homomorphisms of modules
1.2.5 The composition of homomorphisms
1.2.6 The opposite of a ring
1.2.7 Balanced bimodules
1.2.8 Submodules and generators
1.2.9 Kernel and image
1.2.10 Quotient modules
1.2.12 Images and inverse images
1.2.14 Change of rings
1.2.16 Irreducible modules
1.2.18 Maximal elements in ordered sets
1.2.23 Torsion-free modules and spaces over the field of fractions
Exercises
2 DIRECT SUMS AND SHORT EXACT SEQUENCES
2.1 DIRECT SUMS AND FREE MODULES
2.1.1 Internal direct sums
2.1.2 Examples: vector spaces
2.1.3 Examples: abelian groups
2.1.4 The uniqueness of summands
2.1.5 External direct sums
2.1.6 Standard inclusions and projections
2.1.8 Notation
2.1.9 Idempotents
2.1.11 Infinite direct sums
2.1.13 Remarks
2.1.14 Ordered index sets
2.1.15 The module L^A
2.1.16 The module Fr_R(X)
2.1.17 Left-handed notation
2.1.18 Free generating sets
2.1.19 Free modules
2.1.21 Extending maps
Exercises
2.2 MATRICES, BASES, HOMOMORPHISMS OF FREE MODULES
2.2.1 Bases
2.2.2 Standard bases
2.2.3 Coordinates
2.2.5 Matrices for homomorphisms
2.2.7 Change of basis
2.2.9 Matrices of endomorphisms
2.2.10 Normal forms of matrices
2.2.12 Scalar matrices and endomorphisms
2.2.13 Infinite bases
2.2.14 Free left modules
Exercises
2.3 INVARIANT BASIS NUMBER
2.3.1 Some non-IBN rings
2.3.3 Two non-square invertible matrices
2.3.4 The type
Exercises
2.4 SHORT EXACT SEQUENCES
2.4.1 The definition
2.4.2 Four-term sequences
2.4.3 Short exact sequences
2.4.4 Direct sums and splittings
2.4.6 Dual numbers
2.4.7 The group Ext
2.4.8 Pull-backs and push-outs
2.4.10 Base change for short exact sequences
2.4.11 The direct sum of short exact sequences
Exercises
2.5 PROJECTIVE MODULES
2.5.1 The definition and basic properties
2.5.9 Idempotents and projective modules
2.5.12 A cautionary example
2.5.13 Injective modules
Exercises
2.6 DIRECT PRODUCTS OF RINGS
2.6.1 The definition
2.6.2 Central idempotents
2.6.4 Remarks
2.6.5 An illustration
2.6.6 Modules
2.6.7 Homomorphisms
2.6.10 Historical note
Exercises
3 NOETHERIAN RINGS AND POLYNOMIAL RINGS
3.1 NOETHERIAN RINGS
3.1.1 The Noetherian condition
3.1.5 The ascending chain condition and the maximum condition
3.1.11 Module-finite extensions
Exercises
3.2 SKEW POLYNOMIAL RINGS
3.2.1 The definition
3.2.2 Some endomorphisms
3.2.6 The division algorithm
3.2.7 Euclidean domains
3.2.11 Euclid's algorithm
3.2.12 An example
3.2.13 Inner order and the centre
3.2.15 Ideals
3.2.19 Total division
3.2.22 Unique factorization
3.2.23 Further developments
Exercises
3.3 MODULES OVER SKEW POLYNOMIAL RINGS
3.3.1 Elementary operations
3.3.3 Rank and invariant factors
3.3.4 The structure of modules
3.3.7 Rank and invariant factors for modules
3.3.8 Non-cancellation
3.3.12 Serre's Conjecture
3.3.13 Background and developments
Exercises
4 ARTINIAN RINGS AND MODULES
4.1 ARTINIAN MODULES
4.1.1 The definition
4.1.2 Examples
4.1.3 Fundamental properties
4.1.8 Composition series
4.1.11 Multiplicity
4.1.13 Reducibility
4.1.15 Complete reducibility
4.1.22 Fully invariant submodules
4.1.24 The socle series
Exercises
4.2 ARTINIAN SEMISIMPLE RINGS
4.2.1 Definitions and the statement of the Wedderburn-Artin Theorem
4.2.4 Division rings
4.2.5 Matrix rings
4.2.6 Products of matrix rings
4.2.11 Finishing the proof of the Wedderburn-Artin Theorem
4.2.16 Recapitulation of the argument
Exercises
4.3 ARTINIAN RINGS
4.3.1 The Jacobson radical
4.3.2 Examples
4.3.3 Basic properties
4.3.11 Alternative descriptions of the Jacobson radical
4.3.19 Nilpotent ideals and a characterization of Artinian rings
4.3.22 Semilocal rings
4.3.24 Local rings
4.3.27 Further reading
Exercises
5 DEDEKIND DOMAINS
5.1 DEDEKIND DOMAINS AND INVERTIBLE IDEALS
5.1.1 Prime ideals
5.1.4 Coprime ideals
5.1.7 Fractional ideals
5.1.10 Dedekind domains - the definition
5.1.11 The class group
5.1.13 An exact sequence
5.1.15 Ideal theory in a Dedekind domain
5.1.25 Principal ideal domains
5.1.28 Primes versus irreducibles
Exercises
5.2 ALGEBRAIC INTEGERS
5.2.1 Integers
5.2.6 Quadratic fields
5.2.9 Separability and integral closure
Exercises
5.3 QUADRATIC FIELDS
5.3.1 Factorization in general
5.3.2 Factorization in the quadratic case
5.3.3 Explicit factorizations
5.3.4 A summary
5.3.5 The norm
5.3.9 The Euclidean property
5.3.10 The class group again
5.3.13 The class number
5.3.15 Computations of class groups
5.3.21 Function fields
Exercises
6 MODULES OVER DEDEKIND DOMAINS
6.1 PROJECTIVE MODULES OVER DEDEKIND DOMAINS
6.1.7 The standard form
6.1.11 The noncommutative case
Exercises
6.2 VALUATION RINGS
6.2.1 Valuations
6.2.4 Localization
6.2.9 Uniformizing parameters
6.2.10 The localization as a Euclidean domain
6.2.13 Rank and invariant factors
Exercises
6.3 TORSION MODULES OVER DEDEKIND DOMAINS
6.3.1 Torsion modules
6.3.6 The rank
6.3.7 Primary modules
6.3.10 Elementary divisors
6.3.13 Primary decomposition
6.3.17 Elementary divisors again
6.3.18 Homomorphisms
6.3.21 Alternative de compositions
6.3.22 Homomorphisms again
Exercises
References
Index