本书是P. M. Cohn的经典的三卷集代数教材的修订版第一卷，被广大读者所追捧，公认为学习代数入门教材的杰出代表。本书中涵盖了代数的所有重要结果。读者具备一定的线性代数、群和域知识，对理解本书将更有益。本卷次的目次：集合；群；格点和范畴；环和模；代数；多线性代数；域论；二次型和有序域；赋值论；交换环；无限域扩展。
读者对象：数学专业的广大师生。
Preface
Conventions on Terminology
1.Sets
1.1 Finite， Countable and Uncountable Sets
1，2 Zom''s Lemma and Well—ordered Sets
1.3 Graphs
2.Groups
2.1 Definition and Basic Properties
2.2 Permutation Groups
2.3 The Isomorphism Theorems
2.4 Soluble and Nilpotent Groups
2.5 Commutators
2.6 The Frattini Subgroup and the Fitting Subgroup
3.Lattices and Categories
3.1 Definitions； Modular and Distributive Lattices
3.2 Chain Conditions
3.3 Categories
3.4 Boolean Algebras
4.Rings and Modules
4.1 The Definitions Recalled
4.2 The Category of Modules over a Ring
4.3 Semisimple Modules
4.4 Matrix Rings
4.5 Direct Products of Rings
4.6 Free Modules
4.7 Projective and Injective Modules
4.8 The Tensor Product of Modules
4.9 Duality of Finite Abelian Groups
5.Algebras
5.1 Algebras； Definition and Examples
5.2 The Wedderbum Structure Theorems
5.3 The Radical
5.4 The Tensor Product of Algebras
5.5 The Regular Representation； Norm and Trace
5.6 M6bius Functions
6.Muhilinear Algebra
6.1 Graded Algebras
6.2 Free Algebras and Tensor Algebras
6.3 The Hilbert Series of a Graded Ring or Module
6.4 The Exterior Algebra on a Module
7.Field Theory
7.1 Fields and their Extensions
7.2 Splitting Fields
7.3 The Algebraic Closure of a Field
7.4 Separability
7.5 Automorphisms of Field Extensions
7.6 The Fundamental Theorem of Galois Theory
7.7 Roots of Unity
7.8 Finite Fields
7.9 Primitive Elements； Norm and Trace
7.10 Galois Theory of Equations
7.11 The Solution of Equations by Radicals
8.Quadratic Forms and Ordered Fields
8.1 Inner Product Spaces
8.2 Orthogonal Sums and Diagonalization
8.3 The Orthogonal Group of a Space
8.4 The Clifford Algebra and the Spinor Norm
8.5 Witt''s Cancellation Theorem and the Witt Group of a Field
8.6 Ordered Fields
8.7 The Field of Real Numbers
8.8 Formally Real Fields
8.9 The Witt Ring of a Field
8.10 The Symplectic Group
8.11 Quadratic Forms in Characteristic Two
9.Valuation Theory
9.1 Divisibility and Valuations
9.2 Absolute Values
9.3 The p—adic Numbers
9.4 Integral Elements
9.5 Extension of Valuations
10.Commutative Rings
10.1 Operations on Ideals
10.2 Prime Ideals and Factorization
10.3 Localization
10.4 Noetherian Rings
10.5 Dedekind Domains
10.6 Modules over Dedekind Domains
10.7 Algebraic Equations
10.8 The Primary Decomposition
10.9 Dimension
10.10 The Hilbert Nullstellensatz
11.Infinite Field Extensions
11.1 Abstract Dependence Relations
11.2 Algebraic Dependence
11.3 Simple Transcendental Extensions
11.4 Separable and p—radical Extensions
11.5 Derivations
11.6 Linearly Disjoint Extensions
11.7 Composites of Fields
11.8 Infinite Algebraic Extensions
11.9 Galois Descent
11.10 Kummer Extensions
Bibliography
List of Notations
Author Index
Subject Index
內容試閱：
Condition （c） of Theorem 4.3.4 may also be expressed by saying that any exact sequence 0 → N → M → M／N → 0 splits； in other words， every short exact sequence with middle term M splits.This is equivalent to either of the following conditions on （4.2.2）：
（a）There is an R—homomorphism f''： M → M'' such that ff'' = 1M''； this mapping f'' is called a right inverse or retraction for f.
（b）There is an R—homomorphism g'' ： M" → M such that g''g = 1M"； the mapping g'' is called a left inverse or a section for g.
Clearly （a）， （b） hold when im f is complemented； conversely， in case （a） we have M = im f □ ker f'' and in case （b） M = ker g □ im g''， as is easily verified.
Over a field （even skew） the simple modules are just the one—dimensional vector spaces； since every vector space can be written as a sum of one—dimensional spaces， it follows that over a field every module （= vector space） is semisimple.In particular， this proves the existence of a basis for any vector space， even infinite—dimensional.For if V = □ISi and ui is a generator of S， then {ui}i∈I is a basis of V， as is easily checked.
For an arbitrary ring R the theory of semisimple modules is quite similar to the theory of vector spaces over a field.The main difference is that there may be more than one type of simple module.We shall say that two simple R—modules have the same type if they are isomorphic.A semisimple R—module is called isotypic if it can be written as a sum of simple modules all of the same type.In any R—module M the sum of all simple submodules is called the socle of M； thus the semisimple modules are those that coincide with their socle.The sum of all simple submodules of a given isomorphism type a is called the α—socle of M or a type component in M.
Let M be any R—module； a submodule N of M is said to be fully invariant in M if it admits （i.e.is mapped into itself by） all R—endomorphisms of M.We note that for an R—module M， the set S = EndR（M） is just the centralizer in End（M） of the image of R defining the R—action on M， so a subgroup N of M is a fully invariant submodule iff it admits both R and S.In a semisimple module the fully invariant submodules are easily described： they are the sums of type components.
Theorem 4.：1.7.Let R be a ring and M be an R—module.Then：
（ⅰ）For any type α the α—socle is an isotypic submodule of M containing all simple submodules of type α， and the socle is the direct sum of the α—socles， for different α.
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