Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 58
... kG - module and H be a subgroup in G of finite index n not divisible by char k . Modify the proof of Maschke's Theorem to show the following : If V is semisimple as a kH - module , then V is semisimple as a kG - module . = 1 Solution ...
... kG - module and H be a subgroup in G of finite index n not divisible by char k . Modify the proof of Maschke's Theorem to show the following : If V is semisimple as a kH - module , then V is semisimple as a kG - module . = 1 Solution ...
Page 78
... kG = rad kH . Solution . Even without H being normal in G , we have in general kHrad kG C rad kH , by Exercise 6.3 . Therefore , it suffices to prove that , if H is normal in G , then rad kH C rad kG . Consider any simple left kG - module ...
... kG = rad kH . Solution . Even without H being normal in G , we have in general kHrad kG C rad kH , by Exercise 6.3 . Therefore , it suffices to prove that , if H is normal in G , then rad kH C rad kG . Consider any simple left kG - module ...
Page 79
... module W. By Exercise 6 , kG KH W is a semisimple kG - module , so 0 = a⋅ kGkHW≥Σ¤¿α ; ® kH W = Σx ; & a¿W . • This implies that a ; W = 0 for all i , so a ; Є rad kH . We have now shown that rad kG C kG rad kH , and the reverse ...
... module W. By Exercise 6 , kG KH W is a semisimple kG - module , so 0 = a⋅ kGkHW≥Σ¤¿α ; ® kH W = Σx ; & a¿W . • This implies that a ; W = 0 for all i , so a ; Є rad kH . We have now shown that rad kG C kG rad kH , and the reverse ...
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