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| Artin E., Nesbitt C.J., Thrall R.M. — Rings with Minimum Condition |
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| Предметный указатель |
Algebra, commutator 108
Algebra, cyclic 83
Algebra, division 76
Algebra, domain of integrity in 111
Algebra, exponent of 93
Algebra, regular representation of 24
Algebra, simple 65
Analytic linear functions 61
Analytic linear functions, main theorems on, Lem.7.3D 70
Analytic linear functions, main theorems on, Th.7.1A 61
Annihilators 11
Automorphism, inner 46
Automorphism, of a simple algebra, Cor.7.2D 66
Block 107
Cartan invariants 107
Center 29
Chain condition 12
Commutator, algebra 108
Commutator, of a subring 68
Commutator, of two elements 57
Commutator, theorems on, of a suhalgebra, of a simple ring, Th.7.3C, Th.7.3E, Th.7.3F, Th.7.3H 69 ff.
Congruence 6
Conjugates 66
Crossed product 8l
Crossed product, is a simple algebra, Th.8.4C 82
Crossed product, Kronecker product of, Th.8.5A 86
Decomposition numbers 117
Decomposition of homomorphism 42
Decomposition of left ideals, Th.2.6k 18
Decomposition of representations 26
Decomposition of two-sided ideals, Th.2.6D 19
Decomposition uniqueness of, of spaces 9
Degree of a field over subfield 3
Difference ring 10 see
Dimension, of a matrix 20
Dimension, of a minimal right ideal 37
Dimension, of a vector space 2
Direct sum 1
Direct sum, Cor.1.4D on 5
Division algebra 76
Division algebra, maximal separable subfield of 78
Domain of integrity 111
Exponent of a simple algebra 93
Exponent of a simple algebra, main property of, Th.8.6B 94
Factor set 8l
Factor set, associated 84
Field (means commutative field), algebraic extension of 77
Field (means commutative field), cyclic splitting 82
Field (means commutative field), Galois 117
Field (means commutative field), separable 77
Field (means commutative field), splitting 76
Finiteness conditions, locally finite spaces 53
Finiteness conditions, minimum condition on left ideals 12
Finiteness conditions, minimum condition on subspaces 12
Galois field 117
Galois group 82
Homomorphic image, of rings 10
Homomorphic image, of spaces 6
Homomorphism, addition and multiplication of 39 40
Homomorphism, basic-theorems, Cor.1.5D 7
Homomorphism, basic-theorems, Th.1.5A 6
Homomorphism, basic-theorems, Th.1.5C 7
Homomorphism, decomposition of-relative to a decomposition of space 42
Homomorphism, defined by a matrix and set of "bases, Th.3.1A 21
Homomorphism, into 6
Homomorphism, natural, i.e., a mapping which takes each element into its residue class 100
Homomorphism, of a ring into a set of matrices 23
Homomorphism, of spaces 5
Homomorphism, onto 6
Homomorphism, rank of a set of 115 116
Homomorphism, ring of a ring, Th.5.8B 44
Ideal 2
Ideal, condition for minimal left, Th.5.4A 36
Ideal, criteria for indecomposability of, Ths.9.2C, 9.2E 98
Ideal, dimension of minimal right 37
Ideal, indecomposable left, one that cannot be expressed as the direct sum of two left subideals 97 ff.
Ideal, left 2 10
Ideal, left-of a simple ring, Th.5.3D 34
Ideal, main properties of indecomposable nonnilpotent, Cor.9.2F 98
Ideal, main properties of indecomposable nonnilpotent, Th.9.2G 99
Ideal, main properties of indecomposable nonnilpotent, Th.9.3E 102
Ideal, maximal left 11
Ideal, nilpotent 15
Ideal, nilpotent, Cor.2.AB 16
Ideal, nonnilpotent, Th.2.4A 15
Ideal, null 15
Ideal, primitive (minimal) 15 37
Ideal, principal 18
Ideal, right 2
Ideal, simple 21 35
Ideal, simple left 116 (here means the same as primitive-left ideal)
Ideal, two-sided 2 10
Ideal, two-sided-in Kronecker product, Th.7.1B 62
Ideal, two-sided-of semisimple ring, Lem.4.4C 30
Idempotent, an element such that all of its powers are equal 15
Idempotent, belonging to a block 107
Idempotent, center, Th.4.4A 29
Idempotent, existence of, Th.2.4A 15
Idempotent, generating 33
Idempotent, generator of a left ideal in a semisimple ring 28
Idempotent, induced, Lem.9.8E 115
Idempotent, mutually orthogonal 100
Idempotent, primitive 33
Idempotent, zero (it is assumed through-out that the idempotents discussed are different fromzero) 15 96
Indecomposable, constituents of a representation 26
Indecomposable, main theorem on, space, Th.9.2A 97
Indecomposable, spaces over a ring 96
Inverse, homomorphism ring of a space 41
Inverse, isomorphism 39
Inverse, ring 39
Irreducible, constituents of a representation 26
Irreducible, space 3 11
Irreducible, space over simple ring 35
Isomorphism, inverse 39
Isomorphism, of a k-space, Cor.3.13 22
Isomorphism, of spaces 5
Kernel, for ring homomorphism 10
| Kernel, for space, homomorphism 7
Kronecker product, condition for associativity of 57
Kronecker product, condition for product of two subrings to he a, Th.7.1D 62
Kronecker product, minimum condition in, Th.6.10B 60
Kronecker product, of crossed products, Th.8.5A 86
Kronecker product, of simple ring and its reciprocal (Brauer's Theorem), Th.7.1F 64
Kronecker product, of spaces which are direct sums, Cor.6.9B 59
Kronecker product, of two rings, Th.6.8A 57
Kronecker product, of two spaces relative to a ring 51
Kronecker product, two-sided ideals in, of two rings, Th.7.1B 62
Lector space 7 see
Locally finite spaces 53
Loewy series 102
Loewy series, upper 103
Loewy series, upper, Th.9.4B 104
Loewy series, upper, Th.9.4C 104
Matrices, addition of 20
Matrices, definition of 20
Matrices, dimensions of 20
Matrices, multiplication of 20
Matrices, reducible set of 25
Maximal left ideal 11
Maximum condition 12
Minimum condition 12
Minimum condition, in Kronecker product of two rings 60
Minimum condition, on left ideals 12 ff.
Minimum condition, ring with 13
Minimum condition, ring with, Th.2.2C 14
Minimum condition, space with 13
Minimum condition, space with, Lem.22A 13
Nilpotent, ideal 15
Nilpotent, ring 15
Peirce decomposition 17 19 101
Primary ring, completely primary ring (primitive ring): R is completely primary if it is primary and R-N is a sfield 96
Primitive idempotent 33
Product (of subsets) 1
Product of subrings, condition to be Kronecker product, Th.7.1D 62
Radical 17
Radical, rings with 96 ff.
Radical, Th.2.5A on 17
Reciprocal ring 39
Representation (matrix) 23
Representation (matrix), decomposable 26
Representation (matrix), equivalent 24
Representation (matrix), faithful 24 40
Representation (matrix), induced modular 113
Representation (matrix), modular 111
Representation (matrix), reducible 25
Representation (matrix), regular 24
Representation space 24 see
Representation space, decomposable 25
Representation space, reducible 25
Representation space, Th.3.2B on 24
Residue class 6
Residue class, ring 10
Residue class, space 6
Ring, center of a 29
Ring, completely primary 96
Ring, homomorphisms of a, Th.5.8B 44
Ring, inverse (or reciprocal) 39
Ring, inverse homomorphism, of a space 41
Ring, of integers of an algebraic number field 111
Ring, semisimple 27
Ring, simple 27
Ring, with minimum condition on left ideals 12 13
Ring, with radicals 96 ff.
Semisimple ring 27
Semisimple ring, decomposition of a into left ideals, Th.5.3A 33
Semisimple ring, lemmas on, 4.4C 30
Semisimple ring, lemmas on, 4.5A 30
Semisimple ring, main theorem on, Th.4.1A 27
Semisimple ring, space over a, Th.5.3H 36
Sfield component of a simple ring (the sfield given by Wedderburn's Theorem) 76
Sfield division algebra is a 76
Sfield R-homomorphisms of an irreducible space 39
Sfield: A system with all the properties of a field except the necessity of commutativity, variously called in the literature division ring, skew-field, schiefkoerper, quasi-field 1
Simple algebra, group property of similar, Th.8.2D 75
Simple algebra, is similar to a crossed product, Th.8.4B 82
Simple algebra, isomorphism of subalgebras leaving center fixed can be extended to inner automorphism, Th.7.2C 66
Simple ideal 27
Simple ring 27
Simple ring, Kronecker product of a, and its inverse (Brauer's Theorem) 64
Simple ring, main theorem on 32
Simple ring, Similar 75
Simple ring, sum of distinct, is direct 30
Space see “Vector space” “We
Splitting field 76
Splitting field, cyclic 82
Splitting field, existence of, Cor.8.3B 77
Splitting field, main theorem on, 8.3A 76
Subspace (means subspace invariant under multiplication by ring elements) 3
Subspace (means subspace invariant under multiplication by ring elements), ffllnlTrrmn condition on 12
Subspace (means subspace invariant under multiplication by ring elements), irreducible 3
Sum (of subsets) 1
Sum (of subsets) direct 1
Uniqueness, Uniqueness of decomposition of spaces 9
Unit, of a semisimple ring, Th.4.3A 29
Unit, right 28
Unit, rings with 96
Vector space, completely reducible 5 103
Vector space, composition spaces of a 105
Vector space, criterion for indecomposa'bility of a, Oh.9.2A 97
Vector space, dimension of a 2
Vector space, indecomposable 96
Vector space, irreducible 3
Vector space, irreducible, Cor.1.5B 6
Vector space, irreducible, Lem. 1.4A 4
Vector space, irreducible, over simple ring, Th.5.3E 35
Vector space, isomorphic 5
Vector space, Kronecker product of, relative to a ring 51
Vector space, locally finite 53
Vector space, over a semisimple ring, Th.5.3H 36
Vector space, universal models for, over semisimple rings 47
Vectors, dependent, over a sfield 2
Vectors, independent, over a sfield 2
Vectors, over a ring 4 53
Wedderburn's Theorem 32
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