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 |
À l'intérieur du livre
Résultats 1-5 sur 74
Page
... idempotents, and ending with perfect and semiperfect rings. For the reader's information, we should note that this book does not include problems in the vast areas of module theory (e.g., projectivity, injectivity, and flatness) ...
... idempotents, and ending with perfect and semiperfect rings. For the reader's information, we should note that this book does not include problems in the vast areas of module theory (e.g., projectivity, injectivity, and flatness) ...
Page
... Idempotents 267 §19 . Local Rings 267 17 Exercises §20 . Semilocal Rings . .. 278 20 Exercises §21 . The Theory of Idempotents .. 291 35 Exercises §22 . Central Idempotents and Block Decompositions .. 315 8 Perfect and Semiperfect Rings ...
... Idempotents 267 §19 . Local Rings 267 17 Exercises §20 . Semilocal Rings . .. 278 20 Exercises §21 . The Theory of Idempotents .. 291 35 Exercises §22 . Central Idempotents and Block Decompositions .. 315 8 Perfect and Semiperfect Rings ...
Page 4
... idempotents ( resp . central idempotents ) → · e1 , ... , en with sum 1 such that e¿ej O whenever i ‡ j , and Bi = = Re ; for all i . In the case where the B2's are ideals , if R = B1 → → Bn , then each B1 is a ring with identity e ...
... idempotents ( resp . central idempotents ) → · e1 , ... , en with sum 1 such that e¿ej O whenever i ‡ j , and Bi = = Re ; for all i . In the case where the B2's are ideals , if R = B1 → → Bn , then each B1 is a ring with identity e ...
Page 24
... idempotent ) , but since c is nilpotent , a + bx = ( 1 + cx ) a Є U ( A ) · U ( R ) = U ( A ) . 66 For a counterexample to “ — ” , let ƒ = a + cx , where a and c are as above . We claim that ƒ ¢ U ( A ) . To see this , assume there ...
... idempotent ) , but since c is nilpotent , a + bx = ( 1 + cx ) a Є U ( A ) · U ( R ) = U ( A ) . 66 For a counterexample to “ — ” , let ƒ = a + cx , where a and c are as above . We claim that ƒ ¢ U ( A ) . To see this , assume there ...
Page 25
... idempotent e ∈ R, we have the following two distributive laws: (1) eR∩ (A + B)=(eR ∩ A)+(eR ∩B), and (2) eR + (A∩ B)=(eR A) ∩ (eR + + B). (3) Show that (1) and (2) no longer hold if eR is replaced by Re. Solution. (1) We need only ...
... idempotent e ∈ R, we have the following two distributive laws: (1) eR∩ (A + B)=(eR ∩ A)+(eR ∩B), and (2) eR + (A∩ B)=(eR A) ∩ (eR + + B). (3) Show that (1) and (2) no longer hold if eR is replaced by Re. Solution. (1) We need only ...
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 |
Autres éditions - Tout afficher
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