An Introduction to Rings and Modules with K.-theory in view

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