Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 97
... hence so is V ' : = see that VV ' . ( b2 ) Using the same type of calculations as above , we can verify the fol- lowing two properties of M = ke1 → ke2 → ke3 : ( A ) M has no kG - submodule U_ . ( B ) The only kG - submodule of M ...
... hence so is V ' : = see that VV ' . ( b2 ) Using the same type of calculations as above , we can verify the fol- lowing two properties of M = ke1 → ke2 → ke3 : ( A ) M has no kG - submodule U_ . ( B ) The only kG - submodule of M ...
Page 198
... hence n - 2 ) . By our " Initial Observation " above , ƒ is a sum of squares in k ( t ) , so by the inductive hypothesis , ƒ € T. Assume , for the moment , that m > 0 . Write hi Pif + r ; where pi , ri Є R , with deg r ; < m . If all r1 ...
... hence n - 2 ) . By our " Initial Observation " above , ƒ is a sum of squares in k ( t ) , so by the inductive hypothesis , ƒ € T. Assume , for the moment , that m > 0 . Write hi Pif + r ; where pi , ri Є R , with deg r ; < m . If all r1 ...
Page 225
... hence an isomorphism . = ( 3 ) ⇒ ( 2 ) Fix an isomorphism : R " R " M and let π : R " M → R " be the first - component projection onto R " . Then πy is an epimorphism and hence an isomorphism . This implies that π is also an ...
... hence an isomorphism . = ( 3 ) ⇒ ( 2 ) Fix an isomorphism : R " R " M and let π : R " M → R " be the first - component projection onto R " . Then πy is an epimorphism and hence an isomorphism . This implies that π is also an ...
Table des matières
2 Semisimplicity | 16 |
Jacobson Radical Theory | 35 |
5 Jacobson radical under change of rings | 52 |
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 linear local ring M₁ matrix maximal ideal maximal left ideal maximal subfield minimal left Mn(R 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