Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 29
... zero , and ( a ;; ) be an m x m skew symmetric matrix over k . Let R be the k - algebra generated by 1 , ... , Im ... zero block of size t ≥ 0. If t > 0 , then det ( A ) = 0 , and m generates a proper ideal in R. If t = 0 , then det ( A ) ...
... zero , and ( a ;; ) be an m x m skew symmetric matrix over k . Let R be the k - algebra generated by 1 , ... , Im ... zero block of size t ≥ 0. If t > 0 , then det ( A ) = 0 , and m generates a proper ideal in R. If t = 0 , then det ( A ) ...
Page 106
... zero diagonal , and The prime ideals of R are ( pZ Z Z R / rad R Zx Z × Z. 0 Z Z " ( 3 Ꮓ Z Z Z Ꮓ 0 pZ Z 0 Z Z " 0 0 Z 0 0 pZ 0 0 Z where p is zero or a prime number , and the semiprime ideals of R are nj Z Z ( " 0 n2Z 0 0 Z Z nзZ ...
... zero diagonal , and The prime ideals of R are ( pZ Z Z R / rad R Zx Z × Z. 0 Z Z " ( 3 Ꮓ Z Z Z Ꮓ 0 pZ Z 0 Z Z " 0 0 Z 0 0 pZ 0 0 Z where p is zero or a prime number , and the semiprime ideals of R are nj Z Z ( " 0 n2Z 0 0 Z Z nзZ ...
Page 113
... zero iff I is a union of left ideals of square zero . However , I itself need not be nilpotent , even if R is commutative . For instance , let R be the commutative F2 - algebra generated by x1 , x2 , ... with the relations = ... = 0 . x ...
... zero iff I is a union of left ideals of square zero . However , I itself need not be nilpotent , even if R is commutative . For instance , let R be the commutative F2 - algebra generated by x1 , x2 , ... with the relations = ... = 0 . x ...
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