Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 118
... follows that Nil * ( M2 ( R ) ) = Mn ( I ′ ) ≤ Mn ( I ) , so equality holds . ( 3 ) ( 2 ) is clear since any nil ideal I lies in Nil * ( R ) . ( 2 ) ( 4 ) follows from Exercise 24 . = ( 4 ) ( 5 ) The inclusion " C " in ( 5 ) follows ...
... follows that Nil * ( M2 ( R ) ) = Mn ( I ′ ) ≤ Mn ( I ) , so equality holds . ( 3 ) ( 2 ) is clear since any nil ideal I lies in Nil * ( R ) . ( 2 ) ( 4 ) follows from Exercise 24 . = ( 4 ) ( 5 ) The inclusion " C " in ( 5 ) follows ...
Page 267
... follows that ( 1 ) - fifin PiPin → M E is onto for suitable indices i1 , ... , in Є I. To simplify the notation , let us write 1 , ... , n for i1 , ... , in in the following . From ( 1 ) , we have an epimor- phism ( 2 ) n i = 1 ( P ) ...
... follows that ( 1 ) - fifin PiPin → M E is onto for suitable indices i1 , ... , in Є I. To simplify the notation , let us write 1 , ... , n for i1 , ... , in in the following . From ( 1 ) , we have an epimor- phism ( 2 ) n i = 1 ( P ) ...
Page 268
... follows that g € U ( E ) , and in particular M = P , as desired . We have now proved the equivalence of ( 1 ) − ( 5 ) . ( 6 ) ( 5 ) is clear , since Endя ( eR ) ≈ eRe by FC- ( 21.7 ) . ( 1 ) = aß ( 6 ) Let a PS be a projective cover ...
... follows that g € U ( E ) , and in particular M = P , as desired . We have now proved the equivalence of ( 1 ) − ( 5 ) . ( 6 ) ( 5 ) is clear , since Endя ( eR ) ≈ eRe by FC- ( 21.7 ) . ( 1 ) = aß ( 6 ) Let a PS be a projective cover ...
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