Exercises in Classical Ring TheorySpringer Science & Business Media, 9 mai 2006 - 364 pages This useful book, which grew out of the author's lectures at Berkeley, presents some 400 exercises of varying degrees of difficulty in classical ring theory, together with complete solutions, background information, historical commentary, bibliographic details, and indications of possible improvements or generalizations. The book should be especially helpful to graduate students as a model of the problem-solving process and an illustration of the applications of different theorems in ring theory. The author also discusses "the folklore of the subject: the `tricks of the trade' in ring theory, which are well known to the experts in the field but may not be familiar to others, and for which there is usually no good reference". The problems are from the following areas: the Wedderburn-Artin theory of semisimple rings, the Jacobson radical, representation theory of groups and algebras, (semi)prime rings, (semi)primitive rings, division rings, ordered rings, (semi)local rings, the theory of idempotents, and (semi)perfect rings. Problems in the areas of module theory, category theory, and rings of quotients are not included, since they will appear in a later book. T. W. Hungerford, Mathematical Reviews |
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Résultats 6-10 sur 86
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... ring theory, starting with the Wedderburn-Artin theory of semisimple rings, Jacobson's theory of the radical, and the representation theory of groups and algebras, then continuing with prime and semiprime rings, primitive and semiprim- ...
... ring theory, starting with the Wedderburn-Artin theory of semisimple rings, Jacobson's theory of the radical, and the representation theory of groups and algebras, then continuing with prime and semiprime rings, primitive and semiprim- ...
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... Rings 247 15 Exercises §18 . Ordered Division Rings .. 258 7 Exercises 7 Local Rings , Semilocal Rings , and Idempotents 267 §19 . Local Rings 267 17 Exercises §20 . Semilocal Rings . .. 278 20 Exercises §21 . The Theory of Idempotents ...
... Rings 247 15 Exercises §18 . Ordered Division Rings .. 258 7 Exercises 7 Local Rings , Semilocal Rings , and Idempotents 267 §19 . Local Rings 267 17 Exercises §20 . Semilocal Rings . .. 278 20 Exercises §21 . The Theory of Idempotents ...
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... ring theory conventions used in this book are the same as those introduced in FC . Thus , a ring R means a ring with identity ( unless other- wise specified ) . A subring of R means a subring containing the identity of R ( unless ...
... ring theory conventions used in this book are the same as those introduced in FC . Thus , a ring R means a ring with identity ( unless other- wise specified ) . A subring of R means a subring containing the identity of R ( unless ...
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... ring. Some of the exercises in this section lie at the foundations of noncom- mutative ring theory, and will be used freely in all later exercises. These include, for instance, the computation of the center of a matrix ring (Exer- cise ...
... ring. Some of the exercises in this section lie at the foundations of noncom- mutative ring theory, and will be used freely in all later exercises. These include, for instance, the computation of the center of a matrix ring (Exer- cise ...
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... ring Bi . Solution . Let I = In B1 , and write B1 = Re ; as in Exercise 1.7 . We claim that ICI is an equality ... ring R , the center of the matrix ring Mn ( R ) consists of the diagonal matrices r · In , where r belongs to the ...
... ring Bi . Solution . Let I = In B1 , and write B1 = Re ; as in Exercise 1.7 . We claim that ICI is an equality ... ring R , the center of the matrix ring Mn ( R ) consists of the diagonal matrices r · In , where r belongs to the ...
Table des matières
2 | 49 |
6 Group Rings and the JSemisimplicity Problem | 80 |
Introduction to Representation Theory | 132 |
Prime and Primitive Rings | 141 |
3 | 157 |
11 Structure of Primitive Rings the Density Theorem | 161 |
26 | 171 |
32 | 191 |
15 Tensor Products and Maximal Subfields | 228 |
Ordered Structures in Rings 247 | 246 |
17 Orderings and Preorderings in Rings 247 | 253 |
Local Rings Semilocal Rings and Idempotents | 268 |
Perfect and Semiperfect Rings 325 | 326 |
24 Homological Characterizations of Perfect and Semiperfect | 336 |
Name Index | 349 |
49 | 350 |
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Expressions et termes fréquents
0-divisor 2-primal abelian artinian ring assume central idempotents commutative ring conjugate constructed contradiction cyclic defined direct product division algebra division ring domain element endomorphism equation Exercise exists fact FC-p field finite finite-dimensional follows group G hence homomorphism idempotent identity implies indecomposable induction integer inverse isomorphism J-semisimple Jacobson k-algebra kG-module left ideal left primitive ring Lemma local ring Math matrix ring maximal ideal maximal left ideal maximal subfield Mn(R module multiplication Neumann regular ring nil ideal nilpotent ideal noetherian ring noncommutative polynomial prime ideal primitive idempotents primitive rings projective proof prove R-module R/rad radical resp right ideal right perfect right R-module ring theory semilocal ring semiprimary ring semisimple ring simple ring Solution stable range strongly regular subdirect product subdirectly irreducible subgroup submodule subring suffices to show Theorem unit-regular von Neumann regular zero