Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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... module Σm ; E , and we are done . To show that m ; E is a simple E - module , it suffices to check that , for any e Є E such that m ; e # 0 , m ; eЕ contains m ;. Consider the R ... R - module complement of Rm.e. ) Now m ; ef = ( m ; e ) = ( ...
... module Σm ; E , and we are done . To show that m ; E is a simple E - module , it suffices to check that , for any e Є E such that m ; e # 0 , m ; eЕ contains m ;. Consider the R ... R - module complement of Rm.e. ) Now m ; ef = ( m ; e ) = ( ...
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Tsit-Yuen Lam. m that ā ? TM € [ A¡ , A¡ ] for some m ) . Therefore , we may as well assume that R = Mn ( k ) . Here ... module V. But by FC- ( 7.13 ) ( 1 ) , V is a composition factor of MK for some simple left R - module M. Since ( rad ...
Tsit-Yuen Lam. m that ā ? TM € [ A¡ , A¡ ] for some m ) . Therefore , we may as well assume that R = Mn ( k ) . Here ... module V. But by FC- ( 7.13 ) ( 1 ) , V is a composition factor of MK for some simple left R - module M. Since ( rad ...
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Tsit-Yuen Lam. Ex . 24.7 . For any ring R , show that the following are equivalent : ( 1 ) R is right perfect ; ( 2 ) R is semilocal and every right R - module M 0 has a maximal sub- module ; ( 3 ) R is semilocal and every left module N 0 ...
Tsit-Yuen Lam. Ex . 24.7 . For any ring R , show that the following are equivalent : ( 1 ) R is right perfect ; ( 2 ) R is semilocal and every right R - module M 0 has a maximal sub- module ; ( 3 ) R is semilocal and every left module N 0 ...
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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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