Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 50
... minimal left ideals of R ) is an ideal of R. Using this , give a new proof for the fact that if R is a simple ring which has a minimal left ideal , then R is a semisimple ring . Solution . For any minimal left ideal I CR and any rЄ R ...
... minimal left ideals of R ) is an ideal of R. Using this , give a new proof for the fact that if R is a simple ring which has a minimal left ideal , then R is a semisimple ring . Solution . For any minimal left ideal I CR and any rЄ R ...
Page 113
... minimal prime ideal . Using this , show that the lower nilradical Nil✩R is the intersection of all the minimal prime ideals of R. Solution . The second conclusion follows directly from the first , since Nil⭑R is the intersection of ...
... minimal prime ideal . Using this , show that the lower nilradical Nil✩R is the intersection of all the minimal prime ideals of R. Solution . The second conclusion follows directly from the first , since Nil⭑R is the intersection of ...
Page 132
... minimal left ideal of R iff A = Re where e Є R has rank 1 , and that ( 2 ) BCR is a minimal right ideal of R iff BeR where e € R has rank 1 . Solution . ( 1 ) Let A = Re , where eV = vk , v 0. To check that A is a minimal left ideal ...
... minimal left ideal of R iff A = Re where e Є R has rank 1 , and that ( 2 ) BCR is a minimal right ideal of R iff BeR where e € R has rank 1 . Solution . ( 1 ) Let A = Re , where eV = vk , v 0. To check that A is a minimal left ideal ...
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