Algebras, Rings and Modules, Volume 1Springer Science & Business Media, 1 de out. de 2004 - 380 páginas Accosiative rings and algebras are very interesting algebraic structures. In a strict sense, the theory of algebras (in particular, noncommutative algebras) originated fromasingleexample,namelythequaternions,createdbySirWilliamR.Hamilton in1843. Thiswasthe?rstexampleofanoncommutative”numbersystem”. During thenextfortyyearsmathematiciansintroducedotherexamplesofnoncommutative algebras, began to bring some order into them and to single out certain types of algebras for special attention. Thus, low-dimensional algebras, division algebras, and commutative algebras, were classi?ed and characterized. The ?rst complete results in the structure theory of associative algebras over the real and complex ?elds were obtained by T.Molien, E.Cartan and G.Frobenius. Modern ring theory began when J.H.Wedderburn proved his celebrated cl- si?cation theorem for ?nite dimensional semisimple algebras over arbitrary ?elds. Twenty years later, E.Artin proved a structure theorem for rings satisfying both the ascending and descending chain condition which generalized Wedderburn structure theorem. The Wedderburn-Artin theorem has since become a corn- stone of noncommutative ring theory. The purpose of this book is to introduce the subject of the structure theory of associative rings. This book is addressed to a reader who wishes to learn this topic from the beginning to research level. We have tried to write a self-contained book which is intended to be a modern textbook on the structure theory of associative rings and related structures and will be accessible for independent study. |
Conteúdo
1 Preliminaries | 1 |
12 MODULES AND HOMOMORPHISMS | 15 |
13 CLASSICAL ISOMORPHISM THEOREMS | 18 |
14 DIRECT SUMS AND PRODUCTS | 21 |
15 FINITELY GENERATED AND FREE MODULES | 24 |
16 NOTES AND REFERENCES | 27 |
2 Decompositions of rings | 30 |
22 THE WEDDERBURNARTIN THEOREM | 33 |
82 DEDEKIND DOMAINS | 193 |
83 HEREDITARY DOMAINS | 199 |
84 DISCRETE VALUATION RINGS | 201 |
85 FINITELY GENERATED MODULES OVER DEDEKIND DOMAINS | 205 |
86 PRUFER RINGS | 208 |
87 NOTES AND REFERENCES | 209 |
9 Goldie rings | 210 |
92 PRIME AND SEMIPRIME RINGS | 214 |
23 LATTICES BOOLEAN ALGEBRAS AND RINGS | 37 |
24 FINITELY DECOMPOSABLE RINGS | 50 |
25 NOTES AND REFERENCES | 57 |
3 Artinian and Noetherian rings | 59 |
32 THE JORDANHOLDER THEOREM | 64 |
33 THE HILBERT BASIS THEOREM | 67 |
34 THE RADICAL OF A MODULE AND A RING | 68 |
35 THE RADICAL OF ARTINIAN RINGS | 71 |
36 A CRITERION FOR A RING TO BE ARTINIAN OR NOETHERIAN | 74 |
37 SEMIPRIMARY RINGS | 76 |
38 NOTES AND REFERENCES | 77 |
4 Categories and functors | 82 |
42 EXACT SEQUENCES DIRECT SUMS AND DIRECT PRODUCTS | 85 |
43 THE HOM FUNCTORS | 90 |
44 BIMODULES | 93 |
45 TENSOR PRODUCTS OF MODULES | 94 |
46 TENSOR PRODUCT FUNCTOR | 99 |
47 DIRECT AND INVERSE LIMITS | 102 |
48 NOTES AND REFERENCES | 109 |
5 Projectives injectives and flats | 111 |
52 INJECTIVE MODULES | 115 |
53 ESSENTIAL EXTENSIONS AND INJECTIVE HULLS | 125 |
54 FLAT MODULES | 131 |
55 RIGHT HEREDITARY AND RIGHT SEMIHEREDITARY RINGS | 135 |
56 HERSTEINSMALL RINGS | 139 |
57 NOTES AND REFERENCES | 141 |
6 Homological dimensions | 143 |
62 PROJECTIVE AND INJECTIVE RESOLUTIONS DERIVED FUNCTORS | 146 |
63 THE FUNCTOR TOR | 150 |
64 THE FUNCTOR EXT | 153 |
65 PROJECTIVE AND INJECTIVE DIMENSIONS | 155 |
66 GLOBAL DIMENSIONS | 158 |
67 NOTES AND REFERENCES | 159 |
7 Integral domains | 161 |
72 FACTORIAL RINGS | 164 |
73 EUCLIDEAN DOMAINS | 169 |
74 RINGS OF FRACTIONS AND QUOTIENT FIELDS | 171 |
75 POLYNOMIAL RINGS OVER FACTORIAL RINGS | 174 |
76 THE CHINESE REMAINDER THEOREM | 177 |
77 SMITH NORMAL FORM OVER A PID | 178 |
78 FINITELY GENERATED MODULES OVER A PID | 181 |
79 THE FROBENIUS THEOREM | 185 |
710 NOTES AND REFERENCES | 187 |
8 Dedekind domains | 189 |
93 GOLDIE RINGS GOLDIES THEOREM | 219 |
94 NOTES AND REFERENCES | 224 |
10 Semiperfect rings | 226 |
102 NONCOMMUTATIVE DISCRETE VALUATION RINGS | 229 |
103 LIFTING IDEMPOTENTS SEMIPERFECT RINGS | 233 |
104 PROJECTIVE COVERS THE KRULLSCHMIDT THEOREM | 237 |
105 PERFECT RINGS | 243 |
106 EQUIVALENT CATEGORIES | 248 |
107 THE MORITA THEOREM | 255 |
108 NOTES AND REFERENCES | 260 |
11 Quivers of rings | 262 |
112 THE PRIME RADICAL | 269 |
113 QUIVERS FINITE DIRECTED GRAPHS | 272 |
114 THE PRIME QUIVER OF A SEMIPERFECT RING | 281 |
115 THE PIERCE QUIVER OF A SEMIPERFECT RING | 285 |
116 DECOMPOSITIONS OF SEMIPERFECT RINGS | 288 |
117 THE PRIME QUIVER OF AN FDDRING | 291 |
118 THE QUIVER ASSOCIATED WITH AN IDEAL | 293 |
119 THE LINK GRAPH OF A SEMIPERFECT RING | 296 |
1110 NOTES AND REFERENCES | 298 |
12 Serial rings and modules | 300 |
122 SEMIPERFECT PRINCIPAL IDEAL RINGS | 302 |
123 SERIAL TWOSIDED NOETHERIAN RINGS | 304 |
124 PROPERTIES OF SERIAL TWOSIDED NOETHERIAN RINGS | 313 |
125 NOTES AND REFERENCES | 316 |
13 Serial rings and their properties | 319 |
132 THE DROZDWARFIELD THEOREM SERIAL RINGS | 323 |
133 MINORS OF SERIAL RIGHT NOETHERIAN RINGS | 325 |
134 STRUCTURE OF SERIAL RIGHT NOETHERIAN RINGS | 330 |
135 SERIAL RIGHT HEREDITARY RINGS SERIAL SEMIPRIME AND RIGHT NOETHERIAN RINGS | 335 |
136 NOTES AND REFERENCES | 339 |
14 Semiperfect semidistributive rings | 341 |
142 REDUCTION THEOREM FOR 5P5DRINGS | 343 |
143 QUIVERS OF SPSDRWGS | 345 |
144 SEMIPRIME SEMIPERFECT RINGS | 347 |
145 RIGHT NOETHERIAN SEMIPRIME 5P5DRINGS | 351 |
146 QUIVERS OF TILED ORDERS | 355 |
147 QUIVERS OF EXPONENT MATRICES | 357 |
148 EXAMPLES | 361 |
149 NOTES AND REFERENCES | 362 |
SUGGESTIONS FOR FURTHER READING | 365 |
369 | |
377 | |
Outras edições - Ver todos
Algebras, Rings and Modules: Volume 1 Michiel Hazewinkel,Nadiya Gubareni,V.V. Kirichenko Visualização parcial - 2006 |
Algebras, Rings and Modules: Non-commutative Algebras and Rings Michiel Hazewinkel,Nadiya M. Gubareni Visualização parcial - 2016 |
Algebras, Rings and Modules: Non-Commutative Algebras and Rings Michiel Hazewinkel Prévia não disponível - 2021 |
Termos e frases comuns
a₁ Abelian group Analogously arrows Artinian ring Boolean algebra called Consider corollary decomposed defined Definition Denote descending chain diagram direct product direct sum discrete valuation ring division ring epimorphism equivalent essential extension exact sequence exists factorial ring factors field finite dimensional finite number finitely generated modules free module Hence homomorphism idempotents identity injective module integer invertible irreducible isomorphism Jacobson radical Kery left A-module left ideals M₁ Math matrix maximal ideal monomorphism morphism multiplication Nakayama's lemma Noetherian ring nonzero element Obviously P₁ Peirce decomposition permutation polynomial Pr(A prime ideal prime radical principal ideal domain projective cover projective module Proof proposition quiver Q(A right A-module right ideal right Noetherian ring ring of fractions semidistributive semiperfect ring semiprime semisimple ring serial ring simple module SPSD-ring submodule sum of pairwise Suppose T-nilpotent theory two-sided ideal two-sided Peirce decomposition uniserial vertex zero ΕΙ
Referências a este livro
Groups, Rings and Group Rings Antonio Giambruno,Cesar Polcino Milies,Sudarshan K. Sehgal Visualização parcial - 2006 |