Index

Abelian group, 20, 71 Algebraically closed field, 46, 97 Alternating group, 32 Ascending chain condition, 112 Associate elements in a domain, 47, 109 Automorphism

of groups, 29 of modules, 70 of rings, 43

Axiom of choice, 10

Basis or free basis canonical or standard for R^{n}, 72, 79 of a module, 78, 83

Bijective or one-to-one correspondence,7 Binary operation, 19 Boolean algebras, 52 Boolean rings, 51

Cancellation law in a group, 20 in a ring, 39

Cartesian product, 2, 11 Cayley’s theorem, 31 Cayley-Hamilton theorem, 66, 98, 125 Center of group, 22 Change of basis, 83 Characteristic of a ring, 50 Characteristic polynomial

of a homomorphism, 85, 95

of a matrix, 66 Chinese remainder theorem, 50, 108 Classical adjoint of a matrix, 63 Cofactor of a matrix, 62 Comaximal ideals, 108, 120 Commutative ring, 37 Complex numbers, 1, 40, 46, 47, 97, 104 Conjugate, 64 Conjugation by a unit, 44 Contravariant functor, 131 Coproduct or sum of modules, 76 Coset, 24, 42, 74 Cycle, 32 Cyclic

group, 23 module, 107

Determinant of a homomorphism, 85 of a matrix, 60, 128

Diagonal matrix, 56

Dimension of a free module, 83

Division algorithm, 45

Domain euclidean, 116 integral domain, 39 of a function, 5 principal ideal, 46 unique factorization, 111

Dual basis, 132 Dual spaces, 130

Eigenvalues, 95 Eigenvectors, 95 Elementary divisors, 119, 120 Elementary matrices, 58 Elementary operations, 57, 122 Endomorphism of a module, 70 Equivalence class, 4 Equivalence relation, 4 Euclidean algorithm, 14 Euclidean domain, 116 Evaluation map, 47, 49 Even permutation, 32 Exponential of a matrix, 106

Factorization domain (FD), 111 Fermat’s little theorem, 50 Field, 39 Formal power series, 113 Fourier series, 100 Free basis, 72, 78, 79, 83 Free R-module, 78 Function or map, 6

bijective, 7 injective, 7 surjective, 7

Function space Y^{T} as a group, 22, 36 as a module, 69 as a ring, 44 as a set, 12

Fundamental theorem of algebra, 46

Gauss, 113 General linear group GL_{n}(R), 55 Generating sequence in a module, 78 Generators of Z_{n}, 40 Geometry of determinant, 90 Gram-Schmidt orthonormalization, 100 Graph of a function, 6 Greatest common divisor, 15 Group, 19

abelian, 20 additive, 20 cyclic, 23 multiplicative, 19 symmetric, 31

Hausdorff maximality principle, 3, 87,

109 Hilbert, 113 Homogeneous equation, 60 Homormophism of groups, 23 of rings, 42 of modules, 69 Homomorphism of quotient

group, 29 module, 74 ring, 44

Ideal left, 41 maximal, 109 of a ring, 41 prime, 109 principal, 42, 46 right, 41

Idempotent element in a ring, 49, 51 Image of a function, 7 Independent sequence in a module, 78 Index of a subgroup, 25 Index set, 2 Induction, 13 Injective or one-to-one, 7, 79 Inner product spaces, 98 Integers mod n, 27, 40 Integers, 1, 14 Invariant factors, 119 Inverse image, 7 Invertible or non-singular matrix, 55 Irreducible element, 47, 110 Isometries of a square, 26, 34 Isometry, 101 Isomorphism of groups, 29 of modules, 70 of rings, 43

Jacobian matrix, 91 Jordan block, 96, 123 Jordan canonical form, 96, 123, 125

Kernel, 28, 43, 70

Least common multiple, 17, 18 Linear combination, 78 Linear ordering, 3 Linear transformation, 85

Matrix elementary, 58 invertible, 55 representing a linear transformation,

84 triangular, 56

Maximal ideal, 109 independent sequence, 86, 87 monotonic subcollection, 4 subgroup, 114

Minimal polynomial, 127 Minor of a matrix, 62 Module over a ring, 68 Monomial, 48 Monotonic collection of sets, 4 Multilinear forms, 129 Multiplicative group of a finite field, 121

Nilpotent element, 56 homomorphism, 93

Noetherian ring, 112 Normal subgroup, 26

Odd permutation, 32 Onto or surjective, 7, 79 Order of an element or group, 23 Orthogonal group O(n), 102 Orthogonal vectors, 99 Orthonormal sequence, 99

Partial ordering, 3 Partition of a set, 5 Permutation, 31 Pigeonhole principle, 8, 39 Polynomial ring, 45 Power set, 12 Prime

element, 110 ideal, 109 integer, 16

Principal ideal domain (PID), 46 Principal ideal, 42 Product

of groups, 34, 35 of modules, 75 of rings, 49 of sets, 2, 11

Projection maps, 11

Quotient group, 27 Quotient module, 74 Quotient ring, 42

Range of a function, 6 Rank of a matrix, 59, 89 Rational canonical form, 107, 125 Relation, 3 Relatively prime

integers, 16

elements in a PID, 119 Right and left inverses of functions, 10 Ring, 38 Root of a polynomial, 46 Row echelon form, 59

Scalar matrix, 57 Scalar multiplication, 21, 38, 54, 71 Self adjoint, 103, 105 Short exact sequence, 115 Sign of a permutation, 60 Similar matrices, 64 Solutions of equations, 9, 59, 81 Splitting map, 114 Standard basis for R^{n}, 72, 79 Strips (horizontal and vertical), 8 Subgroup, 14, 21 Submodule, 69 Subring, 41 Summand of a module, 77, 115 Surjective or onto, 7, 79 Symmetric groups, 31 Symmetric matrix, 103

Torsion element of a module, 121 Trace

of a homormophism, 85

of a matrix, 65 Transpose of a matrix, 56, 103, 132 Transposition, 32

Unique factorization, in principal ideal domains, 113 of integers, 16

Unique factorization domain (UFD), 111 Unit in a ring, 38

Vector space, 67, 85 Volume preserving homomorphism, 90

Zero divisor in a ring, 39