Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 108
... semiprime , let alone prime . To see that R is prime , view it as a subring of S M2 ( Z ) . Note that nS CR . If a ... semiprime iff , for any two ideals A , B in R , AB = 0 implies that AПB = 0 . ( b ) Let A , B be left ( resp . right ) ...
... semiprime , let alone prime . To see that R is prime , view it as a subring of S M2 ( Z ) . Note that nS CR . If a ... semiprime iff , for any two ideals A , B in R , AB = 0 implies that AПB = 0 . ( b ) Let A , B be left ( resp . right ) ...
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... semiprime iff RK = R & K is semiprime . ( 2 ) Show that Nil * ( RK ) = ( Nil⋆ ( R ) ) K . Solution . ( 1 ) The " if " part is clear ( for any field extension ) , since , for any nilpotent ideal IC R , IK is a nilpotent ideal in RK ...
... semiprime iff RK = R & K is semiprime . ( 2 ) Show that Nil * ( RK ) = ( Nil⋆ ( R ) ) K . Solution . ( 1 ) The " if " part is clear ( for any field extension ) , since , for any nilpotent ideal IC R , IK is a nilpotent ideal in RK ...
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... semiprime ideal ; that is , R / a2R is reduced . This implies that a Є a2R , so we have ( 3 ) . Comment . The fact that we left ( 1 ) ⇒ ( 3 ) open for general rings suggested that it may not hold for noncommutative R. To see that this ...
... semiprime ideal ; that is , R / a2R is reduced . This implies that a Є a2R , so we have ( 3 ) . Comment . The fact that we left ( 1 ) ⇒ ( 3 ) open for general rings suggested that it may not hold for noncommutative R. To see that this ...
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