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| Jacobson N. — Lectures in Abstract Algebra, Vol. 1 |
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| Предметный указатель |
Adjoint of a matrix 59
Algebraic element 94 183
Algebraic extension of a field 101
Algebraic integer 181
Algebraically independent elements 105
Alternating group 37
Alternating group, simplicity of 139
Anti-homomorphism 74
Anti-isomorphism 72
Associated primes of an ideal 174 179
Associates 114
Associative law 8 15
Associative law, generalized 20
Automorphism, group of 45
Automorphism, inner 46
Automorphism, of group 45
Automorphism, of module 165
Automorphism, of ring 68
Binary composition 4
Binary composition, non-associative 18
Binomial theorem 52
Boolean algebra 207
Boolean ring 210
Cayley’s Theorem 28
Center of group 46
Center of ring 64
Chain 188
Chain conditions for groups with operators 142 153 154
Chain conditions for lattices 200
Chain conditions for modules 168
Chain, composition 199
Chain, equivalence of 198
Chain, refinement of 197
Characteristic of ring 74
Characteristic subgroup 130
Circle composition 56
Closure (in a semigroup) 25
Cofactor of a matrix 59
Commutative law 8 21
Commutator 132
Commutator group 132
Complement (in a lattice) 205
Composition, binary 4
Composition, non-associative 18
Composition, ternary 18
Conjugate classes 47
Coset 37
Cover (in a lattice) 188
Decomposable element (in a lattice) 204
Difference ring 66
Dimensionality relation (in modular lattices) 200
Direct product, complete 160
Direct product, of groups with operators 144
Direct product, of invariant subgroups 147
Direct sum 145
Directly projective elements (of a lattice) 204
Distributive law 8
Division ring 54
Divisor (factor) 13 114
Divisor (factor) of ideal 173
Eisenstein’s irreducibility criterion 127
Endomorphism, normal 150
Endomorphism, of group 45
Endomorphism, of module 165
Endomorphism,, radical of 155
Endomorphism,, ring of 80
Endomorphism,, sum of 151
Equivalence classes 5
Equivalence relation 4
Euler  -function 34 67 121
Euler - Fermat theorem 67
Extension of a field 100
Extension of a ring 84
Factor, see Divisor Factor group 41 131
Field 54 183
Field of fractions 88
Field of subsets 207
Field, extension of 100
Field, prime 103
Field, structure of 103
Fitting’s Lemma 155
Fractions 88
Gauss’ Lemma 125
Greatest common divisor 13 118
Greatest common divisor of ideals 173
Group 23
Group of automorphisms 45
Group with operators 128
Group with operators, decomposable 152
Group with operators, determined by a ring 130
Group with operators, direct product of 145
Group with operators, factor group of 131
Group with operators, homogeneous 158
Group with operators, maximal invariant subgroup 140
Group with operators, subgroups of ( -subgroups) 130
Group, cyclic 30
Group, generators of 31
Group, multiplication of 29
Group, regular realizations of 29
Group, simple 139
Group, solvable 139
Hilbert basis theorem 171
Holomorph of group 47
Homomorphism of groups 41
Homomorphism of groups with operators 131
Homomorphism of groups, fundamental theorem for groups with operators 133
Homomorphism of groups, fundamental theorem of 44
Homomorphism of groups, kernel of 43
Homomorphism of groups, natural 43
Homomorphism of lattices 192
Homomorphism of modules 165
Homomorphism of rings 68
Homomorphism of rings, fundamental theorem of 70
Homomorphism of rings, kernel of 69
Ideal 65
Ideal, associated prime 174
Ideal, in a lattice 201
Ideal, left 77
Ideal, primary 174
Ideal, prime 173
Ideal, principal 77
Ideal, radical of 173
Ideal, reducible 175
Ideal, regular 167
Ideal, right 77
Idempotent element 24
Identity element 22
Identity element of lattice 192
Imbedding of commutative inte-gral domain in a field 87
Imbedding of ring in ring with an identity 84
Independence in lattices 202
Induction 7 9
Integers 10
Integers, gaussian 123
Integers, in quadratic fields 184 186
Integral dependence 181
Integral domain 53
Integral domain, Euclidean 122 186
Integral domain, Gaussian 115 184
Integral domain, principal ideal 121
Integrally closed 183 184
Intervals (quotients) in a lattice 192
Intervals (quotients), projective 196
Intervals (quotients), transpose 196
Inverse 22
Irreducible (prime) integer 67
Irreducible element 115
Irreducible element of a lattice 201
Irreducible polynomial 101
Irredundant intersection, of elements of a lattice 201
| Irredundant intersection, of elements of a lattice of ideals 177
Isolated components (of an ideal) 180
Isomorphism theorems for groups with operators 135
Isomorphism, of groups 26
Isomorphism, of lattices 192
Isomorphism, of modules 165
Isomorphism, of rings 68
Jordan - H lder theorem 141
Jordan - Hlder theorem for lattices 199
Krull - Schmidt theorem 156
Kurosch - Ore theorem 204
Lagrange’s Theorem 39
Lattice 189
Lattice principle of duality in 190
Lattice principle of duality in, semi-modular 197
Lattice, complemented 205
Lattice, complete 189
Lattice, composition series in 199
Lattice, distributive 193
Lattice, modular 194
Least common multiple 14 120
Least common multiple of ideals 173
Leibniz’s theorem 100
Length of element of a Gaussian semi-group 116
Length of element of a lattice 199
Linearly ordered set (chain) 188
Mbius function 120
Mapping 3
Mapping, graph of 3
Mapping, induced by an equivalence relation 6
Mapping, inverse 4
Mapping, inverse image of 6
Mapping, order preserving 192
Mapping, resultant of 4
Matrix 56
Matrix, adjoint 59
Matrix, cofactor of 59
Matrix, determinant of 58
Matrix, diagonal 64
Matrix, ring 56
Matrix, scalar 64
Matrix, transposed 72
Maximum condition 169 (see also Chain conditions)
Minimum condition 169
Module 162 163
Module, annihilator 165
Module, cyclic 166
Module, difference 165
Module, generators of 166
Module, modules of a ring 164
Module, quotient 165
Module, unitary 167
Newton’s identities 110
Nilpotent element 55
Order of an element of a group 32
Order of an element of a module 165
Order of semi-group 17
Partially ordered set 187
Peano’s axioms 7
permutations 27
Permutations, decomposition into cycles 34
Permutations, even and odd 36
Poincar ’s theorem 40
Point (in a lattice) 205
Polynomials 93 97
Polynomials, cyclotomic 127
Polynomials, homogeneous 108
Polynomials, in several elements 105
Polynomials, irreducible 101
Polynomials, polynomial functions 111
Polynomials, primitive 124
Polynomials, symmetric 107
Power series 95
Powers 21
Prime element 14 116
Projection 150
Projection, primitive 158
Quadratic extensions of rational field 184
Quasi-regular 55
Quaternions 60
Quaternions, norm of 63
Quaternions, trace of 63
Quotient group see Factor group
Quotient in a lattice 192
Quotient of submodules 165
Radical of ideal 173 175
Realization of a group 28 30
Relation 4
Relation, asymmetry of 9
Relation, reflexivity of 5
Relation, symmetry of 5
Relation, transitivity of 5
Ring 49
Ring, additive group of 50
Ring, boolean 210
Ring, commutative 53
Ring, extension of 84
Ring, group of units of 54
Ring, identity of 53
Ring, multiplications of 82
Ring, multiplicative semi-group of 50
Ring, Noetherian 172
Ring, of formal power series 95
Ring, of polynomials 92
Ring, right annihilator of 82
Ring, simple 70
Schreier’s refinement theorem 138
Schreier’s refinement theorem for lattices 198
Semi-group 15
Semi-group, Gaussian 115
Semi-group, group of units of 25
Semi-group, multiplication table of 17
Semi-group, ring 95
Series, characteristic 143
Series, characteristic, chief 143
Series, characteristic, composition 140 143 199
Series, characteristic, fully invariant 130
Series, characteristic, normal 138
Sets 2
Sets, intersection of 2
Sets, logical sum of 2
Sets, product set 3
Sets, quotient set 5
Stone’s theorem 211
Subdirect product (of groups) 160
Subfield 87
Subgroup 24
Subgroup, characteristic 130
Subgroup, cosets of 37
Subgroup, fully invariant 130
Subgroup, generated by a subset 30
Subgroup, index of 39
Subgroup, invariant (normal) 40
Subgroup, left cosets of 39
Subgroup, products of subgroups 76
Sublattice 191
Submodule 164
Subring 61
Subring, division 63
Subring, generated by a subset 63
Symmetric difference 209
Symmetric group 27
Transcendental element 93
Transcendental extension of a field 101
Transformation group 27
Transformation group, transitive 37
Transformations 4
Transitivity set 37
transpositions 36
Uniqueness of factorization in semigroups 117
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