Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 44
... Neumann regular . Solution . The following are clearly maximal ideals of R : Ma = { ƒ € R : ƒ ( a ) = 0 } , where a Є A. Therefore , rad R ≤ | ma = { ƒ € R : ƒ ( A ) = 0 } = 0 . aЄA To see that in most cases R is not von Neumann regular ...
... Neumann regular . Solution . The following are clearly maximal ideals of R : Ma = { ƒ € R : ƒ ( a ) = 0 } , where a Є A. Therefore , rad R ≤ | ma = { ƒ € R : ƒ ( A ) = 0 } = 0 . aЄA To see that in most cases R is not von Neumann regular ...
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... Neumann regular ring . ( 1 ) Show that the center Z ( R ) is also von Neumann regular . ( 2 ) If R is indecomposable as a ring , then Z ( R ) is a field . Solution . ( 1 ) Let c ER and write c = cxc . Note that we can always " replace ...
... Neumann regular ring . ( 1 ) Show that the center Z ( R ) is also von Neumann regular . ( 2 ) If R is indecomposable as a ring , then Z ( R ) is a field . Solution . ( 1 ) Let c ER and write c = cxc . Note that we can always " replace ...
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... Neumann regular for any i≥ 1. To handle the general case of Sn , choose i such that 2in . To show that ( any ) M E S is regular in Sn , consider the block M 0 matrix ( 생 0 3 ) € S2 . Writing M 0 M 0 0 0 0 0 = A B C D M 0 0 0 ...
... Neumann regular for any i≥ 1. To handle the general case of Sn , choose i such that 2in . To show that ( any ) M E S is regular in Sn , consider the block M 0 matrix ( 생 0 3 ) € S2 . Writing M 0 M 0 0 0 0 0 = A B C D M 0 0 0 ...
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