# Book:N. Bourbaki/Algebra I

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## N. Bourbaki:

## 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*.

### Subject Matter

### 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
- $\map {\operatorname {Hom}_C} {E_1, F_1} \times_C \map {\operatorname {Hom}_C} {E_2, F_2} \to \map {\operatorname {Hom}_C} {E_1 \otimes_C E_2, F_1 \otimes_C F_2}$

- 1. The isomorphisms

- $\S 5$. 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