An Introduction to Rings and Modules: With K-Theory in View (Cambridge Studies in Advanced Mathematics, Series Number 65)

This text is an introduction to the theory of rings and modules for a new graduate student. What gives this book its special flavour is that it is partieularly aimed at someone who intends to move on to study algebraic K-theory This aim influences both the choice of rings and modules that we discuss,. and the manner in which we discuss them. Starting from a knowledge of undergraduate linear algebra together with some elementary properties of the integers, polynomials and matrices, we pro- vide the basic definitions and methods of construction for rings and modules, and then we develop the structure theory for modules over various ldnds of ring. These classes of ring reflect the historical roots of algebraic K-theory hrgeometry and topology on the one hand, and representation theory and number theory on the other. Thus the rings that interest us are Nbetherian. rings, -in particular skew polynomial rings, Artinian rings, and Dedekind domains. The text pursues ring and module theory up to the point where the aspiring K-theorist needs category theory. The necessary category theory is deait with in the companion volume [BK: CM], both in the abstract and in relation ta more advanced topics in the ring and module theory. Our division cf the subject matter in this way means that the present volume can be regarded simply as an introduction to some fundamental topics in the theory -of rings and modules. 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 This concise introduction to ring theory, module theory and number theory is ideal for a first year graduate student, as well as being an excellent reference for working mathematicians in other areas. Starting from definitions, the book introduces fundamental constructions of rings and modules, as direct sums or products, and by exact sequences. It then explores the structure of modules over various types of noncommutative polynomial rings, Artinian rings (both semisimple and not), and Dedekind domains. It also shows how Dedekind domains arise in number theory, and explicitly calculates some rings of integers and their class groups. About 200 exercises complement the text and introduce further topics. This book provides the background material for the authors' forthcoming companion volume Categories and Modules. Armed with these two texts, the reader will be ready for more advanced topics in K-theory, homological algebra and algebraic number theory.