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| Jacobson N. — The Theory of Rings |
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| Предметный указатель |
 -groups 5
 -ring 55
Algebra over a field 55
Algebra over a field, defined by a group 56
Algebra over a field, isomorphism with a matrix algebra 56
Anti-homomorphism of a ring 16
Brauer group of algebras 105
Center of a ring 22 (footnote)
centralizer 103
Chain conditions 7
Chain conditions, rings satisfying descending condition 70
Completely primary ring 57
Completely reducible group 13
Completely reducible group, ring of endomorphisms of 58
Composites of fields 97
Composition series 7
Crossed products 82
Crossed products, direct products of 109
Decomposition of an arbitrary algebra 116
Derivation 101
Direct product of algebras 88
Direct product of crossed products 109
Direct product of fields 96
Direct sums of groups 10
Direct sums of ideals 62
Discriminant of an algebra 113
Division algebras, decomposition into division algebras of prime power degrees 110
Division algebras, degree of 104
Division algebras, over special fields 110
Elementary divisors 44
Elementary matrices 21
Endomorphism of a group 2
Endomorphism of a group, normal 8
Extension of isomorphism between simple subalgebras of a central simple algebra 101
Extension of isomorphism between subfields of a division ring 46
Extension of the field of an algebra 89
Factor set 80
First isomorphism theorem 6
Fitting’s Lemma 9
g-integral 122
Galois theory of division rings 84—86
Groupoid 132
Groupoid of normal ideals 134
Ideals, bounded 38 120
Ideals, capacity of maximal 45 123
Ideals, conjunctive 134
Ideals, distance 134
Ideals, fractional 119
Ideals, integral 120
Ideals, inverse of 120
Ideals, normal 131
Integral closure 122
Invariant factors 49
Involutorial anti-automorphism 24
Jordan — Holder — Schreier theorem 6
Krull — Schmidt theorem 11
Krull — Schmidt theorem, dual of 35
Linear transformation 21
Matrix rings 19 (footnote)
Matrix rings, isomorphism of 25
Minimum polynomial of a matrix (Theorem of Frobenius) 52
Minimum polynomial of an algebra 112
Minimum polynomial of an element of an algebra 110
Modules 14
Modules, bounded 47 129
Modules, conjunctive 137
Modules, cyclic 33
Modules, finitely generated 31 43
Modules, free 32
Modules, left 16
| Modules, order of an element in 33
Nil ring 63
Nilpotent ring 63
Non-commutative polynomial domains 29 35 38 40 46
Norm theorems for division rings 47
Orders 119
Orders, bounded 120
Orders, bounded in an algebra 123
Orders, bounded of an ideal 120
Orders, bounded, equivalent 119
Orders, bounded, factorization of normal ideals in 136
Orders, bounded, factorization of two-sided ideals in 126
Orders, bounded, maximal 121
Primary ring 71
Principal ideal domains 30
Principal ideal domains, embedding in division rings 31
Principal ideal domains, factorization of elements in 34
Principal ideal domains, factorization of two-sided ideals in 37
Principal ideal domains, indecomposable elements in 35
Principal ideal domains, matrices with elements in 41
Principal ideal domains, similarity of elements in 33
Principal ideal rings 75—77
Principal ideal rings, modules relative to 77—79
Principal polynomial, trace and norm 113
Projective representations of a group 80
Quotient ring 118
Radical of a ring of endomorphisms 60
Radical of an abstract ring 64
Real quaternions 111
Real quaternions, algebraic closure of 36
Real quaternions, similarity of matrices over 51
Representation of a ring 14
Representation of a ring, left regular 16
Representation of a ring, regular 15 71—75
Representation of an algebra by matrices 91
Representation of an algebra by matrices with elements in another algebra 94
Representation of an algebra by matrices, complete reducibility for semi-simple algebras 94
Representation space 91
Ring of linear transformations 21
Ring of linear transformations, automorphisms and anti-automorphisms of 23
Ring of linear transformations, commuting subrings of 24
Ring of linear transformations, decomposition into irreducible right ideals 23
Ring of linear transformations, simplicity of 23
Schur’s lemma 57
Second isomorphism theorem 6
Semi-linear transformations 26 49—53
Semi-linear transformations, finite groups of 86
Semi-simple ring 64
Semi-simple ring, representation of 68—70
Semi-simple ring, structure of 64—68
Separability of an algebra 115
Separability of an algebra, discriminant criterion for 116
Similarity of elements 33
Similarity of ideals 130 137 138
Similarity of matrices 50
Simple algebra, central 98
Simple algebra, commuting subalgebras of 103
Simple algebra, exponent of 109
Simple algebra, index of 104
Simple algebra, separable subfields of 106
Simple algebra, simple subalgebras of 101
Simple algebra, structure of (Wedderburn’s Theorem) 98
Simple algebra, subfields of 104
Simple ring 23
Simple ring, structure of 67
Splitting fields 104
Total divisor 40
Vector space 17
Vector space, dimensionality of 18
Vector space, over different division rings 21
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