Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 30
... consider any v ( y ) 0 , say of degree m . Then ( x − p ( y ) ) TM .v ( y ) is a nonzero constant in k . This shows that R v ( y ) = V , so V is simple . Now consider any ƒ Є EndŔV and let ƒ ( 1 ) = g ( y ) . Then , in V = k [ y ] ...
... consider any v ( y ) 0 , say of degree m . Then ( x − p ( y ) ) TM .v ( y ) is a nonzero constant in k . This shows that R v ( y ) = V , so V is simple . Now consider any ƒ Є EndŔV and let ƒ ( 1 ) = g ( y ) . Then , in V = k [ y ] ...
Page 33
... considering the chain of D - subspaces VSV SV 2 .... ( For any subset T C R , TV denotes the D - subspace { Σ¡tivi : t ; Є T , v ; € V } . ) Clearly , the set S CR in ( 1 ) contains 0. Consider all nilpotent subsets S ; CS ( e.g. { 0 ...
... considering the chain of D - subspaces VSV SV 2 .... ( For any subset T C R , TV denotes the D - subspace { Σ¡tivi : t ; Є T , v ; € V } . ) Clearly , the set S CR in ( 1 ) contains 0. Consider all nilpotent subsets S ; CS ( e.g. { 0 ...
Page 116
... consider the upper nilradical Nil * R . If N 2 Nil * R is an ideal with N2 Nil * R , then N is clearly nil , and so N = Nil * R. This checks that Nil * R is semiprime , and we can check the other two properties without difficulty ...
... consider the upper nilradical Nil * R . If N 2 Nil * R is an ideal with N2 Nil * R , then N is clearly nil , and so N = Nil * R. This checks that Nil * R is semiprime , and we can check the other two properties without difficulty ...
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