Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 58
... submodule of V. Following the notations in the proof of FC- ( 6.1 ) , let ƒ V → W be a kH - homomorphism with f | W ... submodule of A. Therefore , A = BOC for a suitable kG - submodule CC A. Going back to the multiplicative notation ...
... submodule of V. Following the notations in the proof of FC- ( 6.1 ) , let ƒ V → W be a kH - homomorphism with f | W ... submodule of A. Therefore , A = BOC for a suitable kG - submodule CC A. Going back to the multiplicative notation ...
Page 97
... submodule U_ . ( B ) The only kG - submodule of M isomorphic to U + is k ( e1 + е2 + е3 ) . We can now show that M is indecomposable . In fact , if otherwise , ( A ) implies that MU + N for some N. Since M has composition factors { U + ...
... submodule U_ . ( B ) The only kG - submodule of M isomorphic to U + is k ( e1 + е2 + е3 ) . We can now show that M is indecomposable . In fact , if otherwise , ( A ) implies that MU + N for some N. Since M has composition factors { U + ...
Page 269
... sub- module ; ( 3 ) R is semilocal and every left module N 0 has a simple submodule . Solution . ( 2 ) or ( 3 ) ⇒ ( 1 ) . Assume ( 2 ) or ( 3 ) . By Exercise 6 , rad R is right T - nilpotent . Since R is semilocal , it follows by ...
... sub- module ; ( 3 ) R is semilocal and every left module N 0 has a simple submodule . Solution . ( 2 ) or ( 3 ) ⇒ ( 1 ) . Assume ( 2 ) or ( 3 ) . By Exercise 6 , rad R is right T - nilpotent . Since R is semilocal , it follows by ...
Table des matières
2 Semisimplicity | 16 |
Jacobson Radical Theory | 35 |
5 Jacobson radical under change of rings | 52 |
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 linear local ring M₁ matrix maximal ideal maximal left ideal maximal subfield minimal left Mn(R 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