# Book:N. Bourbaki/Algebra I

## N. Bourbaki: Algebra I

Published $\text {1968}$, Springer

ISBN 3-540-22525-0.

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

### Contents

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
4. Case of an ordered group of degrees
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
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
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