Jacobson N. — Structure of rings (American mathematical society colloquium publications. Volume XXXVII) :: Электронная библиотека попечительского совета мехмата МГУ

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Jacobson N. — Structure of rings (American mathematical society colloquium publications. Volume XXXVII)

Jacobson N. — Structure of rings (American mathematical society colloquium publications. Volume XXXVII)

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Название: Structure of rings (American mathematical society colloquium publications. Volume XXXVII)

Автор: Jacobson N.

Аннотация:

Since the appearance of the author's Theory of Rings and Artin, Nesbitt and Thrall's Rings with Minimum Condition, a number of important developments have taken place in the theory of (non-commutative) rings. These are: the structure theory of rings without finiteness assumptions, cohomology of algebras, and structure and representation theory of non-semi-simple rings (Frobenius algebras, quasi-Frobenius rings). The main purpose of the present volume is to give an account of the first of these developments. The tools which have been devised for the study of general rings yield improved proofs of the older structure results on rings with minimum condition and of finite dimensional algebras. We have therefore considered the specialization of the general results and methods to these classical cases. Thus the present volume includes virtually all the results on semi-simple rings which can be found in the two books cited before. For example, the theory of centralizers of finite dimensional simple subalgebras of simple rings with minimum condition appears as a special case of the Galois theory of the complete ring of linear transformations of a vector space over a division ling. We believe that the passage to the more general case gives a better insight into these results...

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Рубрика: Математика/

Серия: Lecture Notes in Mathematics

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2, revised

Год издания: 1964

Количество страниц: 307

Добавлена в каталог: 05.12.2010

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Предметный указатель

$\Gamma$ -regular $(\Delta, E)$ -module      172
Adjoint of semi-linear transformation      72
Algebraic algebras      19 210 218 236—237 240—260
Algebraic element      19
Algebras of uniform index      244
Algebras over commutative rings      16
Algebras over commutative rings, over fields      18—20
Annihilator      92
Ascending chain condition      198
Bilinear form      69
Bilinear form, non-degeneracy of      70
Biregular ring      210
Boolean ring      210
Brauer group      120—122 152—156
Cartan-Brauer — Hua theorem      186
Center of semi-simple ring with minimum condition      43
Centralizer of subring      24
Centralizer of subring of completely reducible module      124—126 130—132
Centralizer of subring of irreducible module      26
Centralizer of subring of module      24
Centralizer of subring of right ideal $e \mathfrak{A}$       51
Centralizer of subring of set of linear transformations      32
Centralizer of subring of simple algebra of linear transformations      132
Centralizer of subring of strictly cyclic module      25
Centroid      107
Centroid of simple algebra      108
Closure in finite topology      32
Closure in finite topology of set of primitive ideals      203
Commutativity theorems      183—185 217—222
Completely primary ring      56
Conjugate space      66
Conjugate space, dimension of      68
Conjugate space, total subspaces of      68
Degree of commutativity      228
Density of subdirect sum      15
Density theorems for irreducible modules      28 31
Density theorems for irreducible modules for completely reducible modules      127
Density theorems for irreducible modules for one-sided ideals      33
Derivation      85
Derivation, $(s_{1}, s_{2})$ -derivation      170
Derivation, constants relative to      174
Derivation, extension theorems for      86 151—152
Derivation, higher      171
Descending chain condition for right ideals      38
Differential transformation      87
Dimension of completely reducible module      63
Dimension of completely reducible module of distinguished rings of endomorphisms      138—139
Dimension of completely reducible module of ring relative to subring which is primitive with minimal right ideals      135
Dimension of completely reducible module, left and right dimensions      158 164 175
Dimension of completely reducible module, relative to division subring      157 158
Direct sum complete direct sum of rings      14
Direct sum complete direct sum of rings, discrete direct sum of rings      14
Direct sum complete direct sum of rings, subdirect sum of rings      13—16
Distinguished ring of endomorphisms      127
Distinguished ring of endomorphisms, associated with division subrings      159
Division ring extensions      187—192
Dual vector spaces      69
Enveloping algebra      201
EXPONENT      156
Extension of isomorphisms      150 162
Extension of ring of operators of an algebra      103
Factor set      84 153
Finite Galois theory for division rings      163—165
Fundamental theorem for ring of linear transformation      149
Fundamental theorems for division rings      165 168
Galois $(E, \Delta)$ -modules      142—144
Galois theory Concepts of Galois theory      140—142
Galois theory Concepts of Galois theory for the ring of linear transformations      140—152
Height      135
Ideal of matrix rings      40
Ideal, in semi-simple ring with minimum condition      43
Ideal, maximal      6
Ideal, minimal      50
Ideal, modular      5
Ideal, module isomorphism of right ideals      51
Ideal, nil      8
Ideal, one-sided, in primitive ring with minimal right ideals      90—92
Ideal, order of an element      3
Idempotent elements properties of      48—52
Idempotent elements properties of, central      209
Idempotent elements properties of, completely primitive      59
Idempotent elements properties of, existence of      55 57
Idempotent elements properties of, modulo the radical      53—55
Idempotent elements properties of, orthogonality      49
INDEX      122 135
Index, ring      210
Infinite outer Galois theory for division rings      165—169
Isomorphism theorems for ring of linear transformations      45
Isomorphism theorems for ring of linear transformations for primary rings      59
Isomorphism theorems for ring of linear transformations for primitive ring with minimal right ideals      79
Jordan ring      200
Kronecker product of modules      96
Kronecker product of modules of algebras      102
Kronecker product of modules of algebras of linear transformations      104—107
Kronecker product of modules of distinguished algebras of endomorphisms      132—134
Kronecker product of modules of modules over commutative rings      101
Kronecker product of modules of two-sided modules      99—100
Krull — Schmidt theorem      58
Kurosch’s problem      240—244
Levitzki nil radical      197
Levitzki nil radical, iV-group      140 163
Lie nilpotency      229

Lie ring      85
Lie solvability      229
Linear function      66
Linear transformation of finite rank      74
Linear transformation of finite rank, ideals in ring of      93
Linear transformation of finite rank, rank of      93
Locally nilpotent ring      197
Matrix ring radical of      11
Matrix ring radical of, ideals in      40
Matrix ring radical of, locally matrix ring      237 240
Matrix ring radical of, matrix subrings      237—240 245
Matrix units      52
Maximum condition      198
Minimum condition for right ideals      38
Minimum condition for right ideals for left ideals      42
Minimum polynomial of algebraic element      19
Minimum polynomial of algebraic element, m-sequence      195
Minimum polynomial of algebraic element, m-system      195
Module basic definitions      1—3
Module basic definitions for semi-simple rings with minimum condition      46—47
Module basic definitions of mappings      160
Module basic definitions, completely reducible      47 60—63
Module basic definitions, extension of      83
Module basic definitions, faithful      4
Module basic definitions, irreducibility of two-sided module      17
Module basic definitions, irreducible      4 6—7 24—37
Module basic definitions, irreducible algebra modules      34—35
Module basic definitions, isomorphism of irreducible modules      45
Module basic definitions, kernel of set of modules      4
Module basic definitions, strictly cyclic      5
Module basic definitions, two-sided      17 142—144 169—175
Morse semigroup      198
Multiplication algebra of an algebra      107
Nil radicals definition      193
Nil radicals definition, upper and lower      194
Nil semigroups      198—199 201
Orthogonal complement of a subspace      70
Peirce decompositions      48 50
PI-algebras      222—235
PI-algebras, II-regular ring      210
Primary ring      56
Prime ideal      194
Prime ring      196
Primitive commutative rings      7
Primitive ideal      4
Primitive ideal, topology of set of primitive ideals      203
Primitive ring      4 24—37
Primitive ring, with minimum condition      39
Primitive ring, with non-zero socles      60—94
Product group of modules      95
Quasi-regularity      7
Radical definition      4
Radical definition of polynomial ring      13
Radical definition of related rings      10—13
Radical definition of ring with minimum condition      39
Radical definition, a nil ideal      19—21
Radical definition, characterizations      9—10
Radical definition, under Kronecker multiplication      122—123
Resolvent set      20
Rings of endomorphisms Completely reducible rings of endomorphisms      124—132
SBI-ring      53
Scalar product      80
Schur’s lemma      26
Semi-linear transformation      44
Semi-primary ring      56
Semi-simple ring      4
Semi-simple ring, with minimum condition      40—42
Simple ring      39 108 109 114
Simple ring, characterization of finite dimensional central simple algebras      118
Socle of module      63
Socle of module of ring      64
Socle of module, homogeneous components of      63—64
Spectrum      19
Splitting field      119 122
Standard identities      227
Stone’s theorem      215
Structure space      204
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem)      40
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for algebraic algebras of uniform index      245
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for biregular rings      213
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for Kronecker product of division algebras      176—180
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for Kronecker product of irreducible algebras      113
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for Kronecker product of simple algebras      109 114
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for Kronecker products of primitive algebras having minimal one sided ideads      115—116
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for primary ring      56
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for primitive ring with minimal right ideals      75
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for semi-primary ring      56
Structure theorems for semi-simple ring with minimum condition (Wedderburn-Artin theorem) for simple ring with minimal right ideals      88 90
Subdirectly irreducible rings      219
Subfield separable      178
Subfield separable of division ring      180
Teichmueller’s counter example      147
Topology finite topology      29
Topology finite topology of primitive ring with minimal right ideals      94
Topology finite topology of set of primitive ideals      203
Transitivity, k-fold      32
Uniqueness theorem for direct sum decomposition      42
Uniqueness theorem for direct sum decomposition for structure of primary ring      59
Weakly closed system      200
Weakly Galois subring      142
Wedderburn theorem      183
Wedderburn — Artin theorem      40

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