Book:N. Bourbaki/Algebra I

Originally published as the first $3$ chapters of Éléments de Mathématique II: Algèbre.

Subject Matter

 * Abstract Algebra

Contents

 * To The Reader
 * Contents of the Elements of Mathematics Series


 * Introduction

Chapter I. ALGEBRAIC STRUCTURES

 * $\S 1$. Laws of composition ; associativity; commutativity
 * 1. Laws of composition
 * 2. Composition of an ordered sequence of elements
 * 3. Associative laws
 * 4. Stable subsets. Induced laws
 * 5. Permutable elements. Commutative laws
 * 6. Quotient laws


 * $\S 2$. Identity element; cancellable elements; invertible elements
 * 1. Identity element
 * 2. Cancellable elements
 * 3. Invertible elements
 * 4. Monoid of fractions of a commutative monoid
 * 5. Applications: I. Rational integers
 * 6. Applications: II. Multiplication of rational integers
 * 7. Applications: III. Generalized powers
 * 8. Notation


 * $\S 3$. Actions
 * 1. Actions
 * 2. Subsets stable under an action. Induced action
 * 3. Quotient action
 * 4. Distributivity
 * 5. Distributivity of one internal law with respect to another


 * $\S 4$. Groups and groups with operators
 * 1. Groups
 * 2. Groups with operators
 * 3. Subgroups
 * 4. Quotient groups
 * 5. Decomposition of a homomorphism
 * 6. Subgroups of a quotient group
 * 7. The Jordan-Holder theorem
 * 8. Products and fibre products
 * 9. Restricted sums
 * 10. Monogenous groups


 * $\S 5$. Groups operating on a set
 * 1. Monoid operating on a set
 * 2. Stabilizer, fixer
 * 3. Inner automorphisms
 * 4. Orbits
 * 5. Homogeneous sets
 * 6. Homogeneous principal sets
 * 7. Permutation groups of a finite set


 * $\S 6$. Extensions, solvable groups, nilpotent groups
 * 1. Extensions
 * 2. Commutators
 * 3. Lower central series, nilpotent groups
 * 4. Derived series, solvable groups
 * 5. $p$-groups
 * 6. Sylow subgroups
 * 7. Finite nilpotent groups


 * $\S 7$. Free monoids, free groups
 * 1. Free magmas
 * 2. Free monoids
 * 3. Amalgamated sum of monoids
 * 4. Application to free monoids
 * 5. Free groups
 * 6. Presentations of a group
 * 7. Free commutative groups and monoid
 * 8. Exponential notation
 * 9. Relations between the various free objects


 * $\S 8$. Rings
 * 1. Rings
 * 2. Consequences of distributivity
 * 3. Examples of rings
 * 4. Ring homomorphisms
 * 5. Subrings
 * 6. Ideals
 * 7. Quotient rings
 * 8. Subrings and ideals in a quotient ring
 * 9. Multiplication of ideals
 * 10. Product of rings
 * 11. Direct decomposition of a ring
 * 12. Rings of fractions


 * $\S 9$. Fields
 * 1. Fields
 * 2. Integral domains
 * 3. Prime ideals
 * 4. The field of rational numbers


 * $\S 10$. Inverse and direct limits
 * 1. Inverse systems of magmas
 * 2. Inverse limits of actions
 * 3. Direct systems of magmas
 * 4. Direct limit of actions


 * Exercises for $\S 1$
 * Exercises for $\S 2$
 * Exercises for $\S 3$
 * Exercises for $\S 4$
 * Exercises for $\S 5$
 * Exercises for $\S 6$
 * Exercises for $\S 7$
 * Exercises for $\S 8$
 * Exercises for $\S 9$
 * Exercises for $\S 10$
 * Historical note

Chapter II. LINEAR ALGEBRA

 * $\S 1$. Modules; vector spaces; linear combinations
 * 1. Modules
 * 2. Linear mappings
 * 3. Submodules; quotient modules
 * 4. Exact sequences
 * 5. Products of modules
 * 6. Direct sum of modules
 * 7. Intersection and sum of submodules
 * 8. Direct sums of submodules
 * 9. Supplementary submodules
 * 10. Modules of finite length
 * 11. Free families. Bases
 * 12. Annihilators. Faithful modules. Monogenous modules
 * 13. Change of the ring of scalars
 * 14. Multimodules


 * $\S 2$. Modules of linear mappings. Duality
 * 1. Properties of $\map {\operatorname {Hom}_A } {E, F}$ relative to exact sequences
 * 2. Projective modules
 * 3. Linear forms; dual of a module
 * 4. Orthogonality
 * 5. Transpose of a linear mapping
 * 6. Dual of a quotient module. Dual of a direct sum. Dual bases
 * 7. Bidual
 * 8. Linear equations


 * $\S 3$. Tensor products
 * 1. Tensor product of two modules
 * 2. Tensor product of two linear mappings
 * 3. Change of ring
 * 4. Operators on a tensor product; tensor products as multimodules
 * 5. Tensor product of two modules over a commutative ring
 * 6. Properties of $E \otimes_A F$ relative to exact sequences
 * 7. Tensor products of products and direct sums
 * 8. Associativity of the tensor product
 * 9. Tensor product of families of multimodules


 * $\S 4$. Relations between tensor products and homomorphism modules
 * 1. The isomorphisms
 * $\map {\operatorname {Hom}_B} {E \otimes_A F, G} \to \map {\operatorname {Hom}_A} {F, \map {\operatorname {Hom}_B} {E, G} }$
 * and
 * $\map {\operatorname {Hom}_C} {E \otimes_A F, G} \to \map {\operatorname {Hom}_A} {F, \map {\operatorname {Hom}_C} {E, G} }$
 * 2. The homomorphism $E* \otimes_A F \to \map {\operatorname {Hom}_A} {E, F}$
 * 3. Trace of an endomorphism
 * 4. The homomorphism
 * $\map {\operatorname {Hom}_C} {E_1, F_1} \times_C \map {\operatorname {Hom}_C} {E_2, F_2} \to \map {\operatorname {Hom}_C} {E_1 \otimes_C E_2, F_1 \otimes_C F_2}$


 * $\S 5$. Extension of the ring of scalars
 * 1. Extension of the ring of scalars of a module
 * 2. Relations between restriction and extension of the ring of scalars
 * 3. Extension of the ring of operators of a homomorphism module
 * 4. Dual of a module obtained by extension of scalars
 * 5. A criterion for finiteness


 * $\S 6$. Inverse and direct limits of modules
 * 1 Inverse. limits of modules
 * 2 Direct limits of modules
 * 3 Tensor product of direct limits


 * $\S 7$. Vector spaces
 * 1. Bases of a vector space
 * 2. Dimension of vector spaces
 * 3. Dimension and codimension of a subspace of a vector space
 * 4. Rank of a linear mapping
 * 5. Dual of a vector space
 * 6. Linear equations in vector spaces
 * 7. Tensor product of vector spaces
 * 8. Rank of an element of a tensor product
 * 9. Extension of scalars for a vector space
 * 10. Modules over integral domains


 * $\S 8$. Restriction of the field ofscalars in vector spaces
 * 1. Definition of $K`$-structures
 * 2. Rationality for a subspace
 * 3. Rationality for a linear mapping
 * 4. Rational linear forms
 * 5. Application to linear systems
 * 6. Smallest field of rationality
 * 7. Criteria for rationality


 * $\S 9$. Affine spaces and projective spaces
 * 1. Definition of affine spaces
 * 2. Barycentric calculus
 * 3. Affine linear varieties
 * 4. Affine linear mappings
 * 5. Definition of projective spaces
 * 6. Homogeneous coordinates
 * 7. Projective linear varieties
 * 8. Projective completion of an affine space
 * 9. Extension of rational functions
 * 10. Projective linear mappings
 * 11. Projective space structure


 * $\S 10$. Matrices
 * 1. Definition of matrices
 * 2. Matrices over a commutative group
 * 3. Matrices over a ring
 * 4. Matrices and linear mappings
 * 5. Block products
 * 6. Matrix of a semi-linear mapping
 * 7. Square matrices
 * 8. Change of bases
 * 9. Equivalent matrices ; similar matrices
 * 10. Tensor product of matrices over a commutative ring
 * 11. Trace of a matrix
 * 12. Matrices over a field
 * 13. Equivalence of matrices over a field


 * $\S 11$. Graded modules and rings
 * 1. Graded commutative groups
 * 2. Graded rings and modules
 * 3. Graded submodules
 * 4. Case of an ordered group of degrees
 * 5. Graded tensor product of graded modules
 * 6. Graded modules of graded homomorphisms


 * Appendix. Pseudomodules
 * 1. Adjunction of a unit element to a pseudo-ring
 * 2. Pseudomodules


 * Exercises for $\S 1$
 * Exercises for $\S 2$
 * Exercises for $\S 3$
 * Exercises for $\S 4$
 * Exercises for $\S 5$
 * Exercises for $\S 6$
 * Exercises for $\S 7$
 * Exercises for $\S 8$
 * Exercises for $\S 9$
 * Exercises for $\S 10$
 * Exercises for $\S 11$
 * Exercise for the Appendix

Chapter III. TENSOR ALGEBRAS. EXTERIOR ALGEBRAS. SYMMETRIC ALGEBRAS

 * $\S 1$. Algebras
 * 1. Definition of an algebra
 * 2. Subalgebras. Ideals. Quotient algebras
 * 3. Diagrams expressing associativity and commutativity
 * 4. Products of algebras
 * 5. Restriction and extension of scalars
 * 6. Inverse and direct limits of algebras
 * 7. Bases of an algebra. Multiplication table


 * $\S 2$. Examples of algebras
 * 1. Endomorphism algebras
 * 2. Matrix elements
 * 3. Quadratic algebras
 * 4. Cayley algebras
 * 5. Construction of Cayley algebras. Quaternions
 * 6. Algebra of a magma, a monoid, a group
 * 7. Free algebras
 * 8. Definition of an algebra by generators and relations
 * 9. Polynomial algebras
 * 10. Total algebra of a monoid
 * 11. Formal power series over a commutative ring


 * $\S 3$. Graded algebras
 * 1. Graded algebras
 * 2. Graded subalgebras, graded ideals of a graded algebra
 * 3. Direct limits of graded algebras


 * $\S 4$. Tensor products of algebras
 * 1. Tensor product of a finite family of algebras
 * 2. Universal characterization of tensor products of algebras
 * 3. Modules and multimodules over tensor products of algebras
 * 4. Tensor product of algebras over a field
 * 5. Tensor product of an infinite family of algebras
 * 6. Commutation lemmas
 * 7. Tensor product of graded algebras relative to commutation factors
 * 8. Tensor product of graded algebras of the same types
 * 9 Anticommutative algebras and alternating algebras


 * $\S 5$. Tensor algebra. Tensors
 * 1. Definition of the tensor algebra of a module
 * 2. Functorial properties of the tensor algebra
 * 3. Extension of the ring of scalars
 * 4. Direct limit of tensor algebras
 * 5. Tensor algebra of a direct sum. Tensor algebra of a free module. Tensor algebra of a graded module
 * 6. Tensors and tensor notation


 * $\S 6$. Symmetric algebras
 * 1. Symmetric algebra of a module
 * 2. Functorial properties of the symmetric algebra
 * 3. n-th symmetric power of a module and symmetric multilinear mappings
 * 4. Extension of the ring of scalars
 * 5. Direct limit of symmetric algebras
 * 6. Symmetric algebra of a direct sum. Symmetric algebra of a free module. Symmetric algebra of a graded module


 * $\S 7$. Exterior algebras
 * 1. Definition of the exterior algebra of a module
 * 2. Functorial properties of the exterior algebra
 * 3. Anticommutativity of the exterior algebra
 * 4. n-th exterior power of a module and alternating multilinear mappings
 * 5. Extension of the ring of scalars
 * 6. Direct limits of exterior algebras
 * 7. Exterior algebra of a direct sum. Exterior algebra of a graded module
 * 8. Exterior algebra of a free module
 * 9. Criteria for linear independence


 * $\S 8$. Determinants
 * 1. Determinants of an endomorphism
 * 2. Characterization of automorphisms of a finite-dimensional free module
 * 3. Determinant of a square matrix
 * 4. Calculation of a determinant
 * 5. Minors of a matrix
 * 6. Expansions of a determinant
 * 7. Application to linear equations
 * 8. Case of a commutative field
 * 9. The unimodular group $\SL {n, A}$
 * 10. The $A \sqbrk X$-module associatcd with an $A$-module endomorphism
 * 11. Charactcristic polynoniial of an endomorphism


 * $\S 9$. Norms and traces
 * 1. Norms and traces relative to a module
 * 2. Properties of norms and traces relative to a module
 * 3. Norm and trace in an algebra
 * 4. Properties of norms and traces in an algebra
 * 5. Discriminant of an algebra


 * $\S 10$. Derivations
 * 1. Commutation factors
 * 2. General definition of derivations
 * 3. Examples of derivations
 * 4. Composition of derivations
 * 5. Derivations of an algebra A into an A-module
 * 6. Derivations of an algebra
 * 7. Functorial properties
 * 8. Relations between derivations and algebra homomorphisms
 * 9. Extension of derivations
 * 10. Universal problem for derivations: non-commutative case
 * 11. Universal problem for derivations: commutative case
 * 12. Functorial properties of K-differentials


 * $\S 11$. Cogebras. products of multilinear forms. inner products and duality
 * 1. Cogebras
 * 2. Coassociativity. cocommutativity. counit
 * 3. Properties of graded cogebras of type $\mathbb N$
 * 4 Bigebras and skew-bigebras
 * 5. The graded duals $\map T M^{*gr}$, $\map S M ^{*gr}$ and $\map A M^{*gr}$
 * 6. Inner products: case of algebras
 * 7. Inner products: case of cogebras
 * 8. Inner products: case of bigebras
 * 9. Inner products between $\map T M$ and $\map T {M*}$, $\map S M$ and $\map S {M*}$, $\map \bigwedge M$ and $\map \bigwedge {M*}$
 * 10. Explicit form of inner products in the case of a finitely generated free module
 * 11. Isomorphisms between $\map {\bigwedge^p} M$ and $\map {\bigwedge^{n - p} } {M*}$ for an $n$-dimensional free module $M$
 * 12. Application to the subspace associated with a $p$-vector
 * 13. Pure $p$-vectors. Grassmannians


 * Appendix. Alternative algebras. Octonions
 * 1. Alternative algebras
 * 2. Alternative Cayley algebras
 * 3. Octonions


 * Exercises for $\S 1$
 * Exercises for $\S 2$
 * Exercises for $\S 3$
 * Exercises for $\S 4$
 * Exercises for $\S 5$
 * Exercises for $\S 6$
 * Exercises for $\S 7$
 * Exercises for $\S 8$
 * Exercises for $\S 9$
 * Exercises for $\S 10$
 * Exercises for $\S 11$
 * Exercises for the Appendix


 * INDEX OF NOTATION
 * INDEX OF TERMINOLOGY