Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 43
... Solution . The J - semisimplicity of R / I means that the intersection of the maximal left ideals of R containing I is exactly I. It follows that rad R , the intersection of all the maximal left ideals of R , is contained in I. Ex ...
... Solution . The J - semisimplicity of R / I means that the intersection of the maximal left ideals of R containing I is exactly I. It follows that rad R , the intersection of all the maximal left ideals of R , is contained in I. Ex ...
Page 187
... solution in D. Conversely , if ax in D , then ax - " - - xa = 1 has a solution f ( t ) = h ( t ) ( t − a ) 2 for some h ( t ) € D [ t ] . - Since f ( t ) , ( t − a ) 2 € E [ t ] , Exercise 1 shows that h ( t ) Є E [ t ] . Now f ( t ) ...
... solution in D. Conversely , if ax in D , then ax - " - - xa = 1 has a solution f ( t ) = h ( t ) ( t − a ) 2 for some h ( t ) € D [ t ] . - Since f ( t ) , ( t − a ) 2 € E [ t ] , Exercise 1 shows that h ( t ) Є E [ t ] . Now f ( t ) ...
Page 188
... solution x Є A. Comment . Cohn's formulation of the above result is quite a bit more gen- eral . Let A , B be k - algebras ( where k is as above ) , and let C be an ( A , B ) - bimodule such that ac = ca for every a Є k and c E C. For ...
... solution x Є A. Comment . Cohn's formulation of the above result is quite a bit more gen- eral . Let A , B be k - algebras ( where k is as above ) , and let C be an ( A , B ) - bimodule such that ac = ca for every a Є k and c E C. For ...
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