Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 105
... prime ideal , A be a left ideal and B be a right ideal . Does ABC p imply that A Cp or B C p ? Solution . The answer is " no . " For instance , let R be any prime ring with an idempotent e 0,1 . ( We can take R = Mn ( Z ) with n ≥ 2 ...
... prime ideal , A be a left ideal and B be a right ideal . Does ABC p imply that A Cp or B C p ? Solution . The answer is " no . " For instance , let R be any prime ring with an idempotent e 0,1 . ( We can take R = Mn ( Z ) with n ≥ 2 ...
Page 107
... ideal . Since I IC 12 , we must have IC 12 and hence I = 12 . ( 2 ) ⇒ ( 1 ) Let p be any ideal # R , and let I , Jp be two ideals such that IJ C p . We wish to show that I = p or J = p . By ... Prime Radical ; Prime and Semiprime Rings 107.
... ideal . Since I IC 12 , we must have IC 12 and hence I = 12 . ( 2 ) ⇒ ( 1 ) Let p be any ideal # R , and let I , Jp be two ideals such that IJ C p . We wish to show that I = p or J = p . By ... Prime Radical ; Prime and Semiprime Rings 107.
Page 113
... ideal , and I contains the nonzero nilpotent left ideal Ran - 1 . For ( 2 ) , let J = Nil R. If the image I of I in R / J is nonzero , ( 1 ) would give a nonzero nilpotent ideal in R / J ... Prime Radical ; Prime and Semiprime Rings 113.
... ideal , and I contains the nonzero nilpotent left ideal Ran - 1 . For ( 2 ) , let J = Nil R. If the image I of I in R / J is nonzero , ( 1 ) would give a nonzero nilpotent ideal in R / J ... Prime Radical ; Prime and Semiprime Rings 113.
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