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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... inverse an−1c and a left inverse can−1, so a ∈ U(R). For (b), say ba = 1. Then (ab − 1)a = a − a = 0. If a is not a right 0-divisor, then ab = 1 and so a ∈ U(R). (c) follows immediately from (b). Ex. 1.4*. Let a ∈ R, where R is ...
... inverse an−1c and a left inverse can−1, so a ∈ U(R). For (b), say ba = 1. Then (ab − 1)a = a − a = 0. If a is not a right 0-divisor, then ab = 1 and so a ∈ U(R). (c) follows immediately from (b). Ex. 1.4*. Let a ∈ R, where R is ...
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... inverse b but no left inverse. Show that a has infinitely many right inverses. (In particular, if a ring is finite, it must be Dedekind-finite.) Solution. Suppose we have already constructed n distinct right inverses b1 ,...,b n for a ...
... inverse b but no left inverse. Show that a has infinitely many right inverses. (In particular, if a ring is finite, it must be Dedekind-finite.) Solution. Suppose we have already constructed n distinct right inverses b1 ,...,b n for a ...
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... inverse. Therefore, {b1, bi + c1, ..., bi + cn ) are n + 1 distinct right inverses for a. Alternative Solution. Consider the elements d; = (1 – ba)a' (j > 1), which have the property that adj = 0. We claim that the dj's are distinct ...
... inverse. Therefore, {b1, bi + c1, ..., bi + cn ) are n + 1 distinct right inverses for a. Alternative Solution. Consider the elements d; = (1 – ba)a' (j > 1), which have the property that adj = 0. We claim that the dj's are distinct ...
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... a “conjugate” of the subring R in the ring M2(k). In particular, A = R. Solution. Let O = ( ) . Then o' Ro consists of the matrices (We can define mutually inverse isomorphisms f : R → §1. Basic Terminology and Examples 11.
... a “conjugate” of the subring R in the ring M2(k). In particular, A = R. Solution. Let O = ( ) . Then o' Ro consists of the matrices (We can define mutually inverse isomorphisms f : R → §1. Basic Terminology and Examples 11.
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T.Y. Lam. (We can define mutually inverse isomorphisms f : R → A and g : A → R explicitly by f ( x 0 y z ) = ( x − y x − y − z y y + z ) ∈A, and g ( ) ( a b c d = a + c 0 c d−c ) ∈R, where a + c = b + d.) Comment. A similar ...
T.Y. Lam. (We can define mutually inverse isomorphisms f : R → A and g : A → R explicitly by f ( x 0 y z ) = ( x − y x − y − z y y + z ) ∈A, and g ( ) ( a b c d = a + c 0 c d−c ) ∈R, where a + c = b + d.) Comment. A similar ...
Table des matières
Jacobson Radical Theory | 49 |
Introduction to Representation Theory | 99 |
Ordered Structures in Rings 247 | 246 |
Perfect and Semiperfect Rings 325 | 324 |
Name Index | 349 |
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Expressions et termes fréquents
0-divisor 2-primal abelian algebra artinian ring assume automorphism commutative ring conjugate constructed contradiction cyclic Dedekind-finite defined direct product direct summand division ring domain element endomorphism equation Exercise exists fact field finite group finite-dimensional follows group G hence homomorphism hopfian idempotent identity implies indecomposable induction infinite integer inverse irreducible isomorphism J-semisimple Jacobson radical k-algebra kG-module left ideal left primitive Lemma Let G local ring Math maximal ideal maximal left ideal Mn(k Mn(R module multiplication Neumann regular ring nil ideal Nilº noetherian ring noncommutative nonzero polynomial prime ideal primitive rings proof prove R-module R/rad rad kG representation resp right ideal right R-module ring theory semilocal ring semiprime semisimple ring show that rad simple ring Solution stable range subdirect product subgroup submodule subring suffices to show surjective Theorem unit-regular von Neumann regular zero
Informations bibliographiques
| Titre | Exercises in Classical Ring Theory Problem Books in Mathematics |
| Auteur | T.Y. Lam |
| Édition | 2, illustrée |
| Éditeur | Springer Science & Business Media, 2006 |
| ISBN | 0387217711, 9780387217710 |
| Longueur | 364 pages |
| Exporter la citation | BiBTeX EndNote RefMan |