Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 23
... module . Once this is proved , then m is contained in the semisimple E - module Σm¿E , and we are done . To show that m1E is a simple E - module , it suffices to check that , for any e Є E such that m ; e 0 , mieЕ contains m . Consider ...
... module . Once this is proved , then m is contained in the semisimple E - module Σm¿E , and we are done . To show that m1E is a simple E - module , it suffices to check that , for any e Є E such that m ; e 0 , mieЕ contains m . Consider ...
Page 73
... module M , there exists a ( simple ) left RK - module U such that UL≈ M. Solution . The " only if " part follows directly from FC- ( 7.14 ) . For the " if " part , assume every simple ( left ) RL - module " comes from " a simple RK .
... module M , there exists a ( simple ) left RK - module U such that UL≈ M. Solution . The " only if " part follows directly from FC- ( 7.14 ) . For the " if " part , assume every simple ( left ) RL - module " comes from " a simple RK .
Page 78
... module . Therefore , ( rad kH ) V = 0. By FC- ( 4.1 ) , this implies that rad kH C rad kG . Comment . If H is not ... module and W be a kH - module . Show that ( 1 ) V is a semisimple kG - module iff HV is a semisimple kH - module . ( 2 ) ...
... module . Therefore , ( rad kH ) V = 0. By FC- ( 4.1 ) , this implies that rad kH C rad kG . Comment . If H is not ... module and W be a kH - module . Show that ( 1 ) V is a semisimple kG - module iff HV is a semisimple kH - module . ( 2 ) ...
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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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