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Artin E., Nesbitt C.J., Thrall R.M. — Rings with Minimum Condition
Artin E., Nesbitt C.J., Thrall R.M. — Rings with Minimum Condition



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Название: Rings with Minimum Condition

Авторы: Artin E., Nesbitt C.J., Thrall R.M.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1946

Количество страниц: 123

Добавлена в каталог: 18.04.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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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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