Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 99
... acts irreducibly on V. ) Solution . Consider an irreducible linear group G C GL ( V ) , where V is a finite - dimensional vector space over a field k . We claim that Z ( G ) is cyclic . This will show , for instance , that the Klein 4 ...
... acts irreducibly on V. ) Solution . Consider an irreducible linear group G C GL ( V ) , where V is a finite - dimensional vector space over a field k . We claim that Z ( G ) is cyclic . This will show , for instance , that the Klein 4 ...
Page 128
... acts densely on Vk . By Exercise 7 , the ideal ( x1 , ... , xn ) of R generated by the r2's also acts densely on V. The ring S in question is just A + ( x1 , ... , xn ) . Since S acts densely on Vk , FC- ( 11.20 ) implies that S is a ...
... acts densely on Vk . By Exercise 7 , the ideal ( x1 , ... , xn ) of R generated by the r2's also acts densely on V. The ring S in question is just A + ( x1 , ... , xn ) . Since S acts densely on Vk , FC- ( 11.20 ) implies that S is a ...
Page 130
... acts densely on VK . Ex . 11.15 . For any k - subspace WCV , let ann ( W ) = { r Є R : _rW = 0 } , and , for any left ideal A C R , let ann ( A ) = { v € V : Av = 0 } . Suppose n = dim W∞ . Without assuming the Density Theorem , show ...
... acts densely on VK . Ex . 11.15 . For any k - subspace WCV , let ann ( W ) = { r Є R : _rW = 0 } , and , for any left ideal A C R , let ann ( A ) = { v € V : Av = 0 } . Suppose n = dim W∞ . Without assuming the Density Theorem , show ...
Table des matières
2 Semisimplicity | 16 |
Jacobson Radical Theory 35 1981 | 35 |
6 Group rings and the Jsemisimplicity problem | 57 |
Droits d'auteur | |
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Expressions et termes fréquents
a₁ abelian artinian ring assume automorphism B₁ central idempotents char commutative ring conjugate consider constructed contradiction decomposition Dedekind-finite defined division ring domain element endomorphism equation Exercise exists fact finite group finite-dimensional follows group G hopfian idempotent identity implies indecomposable integer inverse irreducible isomorphism J-semisimple Jacobson radical k-algebra kG-module left ideal left primitive ring Lemma Let G linear local ring M₁ matrix maximal ideal maximal left ideal maximal subfield minimal left module multiplication Neumann regular ring nil ideal nilpotent ideal noetherian ring noncommutative nonzero polynomial prime ideal primitive idempotents primitive rings proof prove quasi-regular R-module R/rad rad kG representation resp right ideal right R-module ring theory semilocal ring semiprime semisimple ring show that rad simple left R-module simple ring soc(RR Solution stable range strongly regular subdirect product subgroup subring Theorem unit-regular zero