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
Page
... matrix units trace ( of a matrix or a field element ) determinant of a matrix cyclic group generated by x center of the group ( or the ring ) G centralizer of A in G H is a normal subgroup of G index of subgroup H in a group G field ...
... matrix units trace ( of a matrix or a field element ) determinant of a matrix cyclic group generated by x center of the group ( or the ring ) G centralizer of A in G H is a normal subgroup of G index of subgroup H in a group G field ...
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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 ... matrix ring Mn ( R ) consists of the diagonal matrices r · In , where r belongs to the center of R. Solution . Let ...
... ring Bi . Solution . Let I = In B1 , and write B1 = Re ; as in Exercise 1.7 . We claim that ICI is an equality ... matrix ring Mn ( R ) consists of the diagonal matrices r · In , where r belongs to the center of R. Solution . Let ...
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... rings with unity,” Amer. Math. Monthly 70 (1963), 315. As for the second solution, a more elaborate construction is possible. Given ab = 1 = ba bi(1 − ba)aj give a in a ring, Jacobson has set of “matrix units” in shown that the ...
... rings with unity,” Amer. Math. Monthly 70 (1963), 315. As for the second solution, a more elaborate construction is possible. Given ab = 1 = ba bi(1 − ba)aj give a in a ring, Jacobson has set of “matrix units” in shown that the ...
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... matrix only if all bk , one . Since is an R - homomorphism and both A Ck = 0. Therefore , is one- and M2 ( C ) have dimension Solution . ( 1 ) Clearly k [ x ]. 8 over R , it follows that is an isomorphism . ( Note that 7 here is not a ...
... matrix only if all bk , one . Since is an R - homomorphism and both A Ck = 0. Therefore , is one- and M2 ( C ) have dimension Solution . ( 1 ) Clearly k [ x ]. 8 over R , it follows that is an isomorphism . ( Note that 7 here is not a ...
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