Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 7
... endomorphism of M is an automorphism . ( 1 ) Show that any noetherian module M is hopfian . ( 2 ) Show that the left regular module RR is hopfian iff R is Dedekind- finite . ( 3 ) Deduce from ( 1 ) , ( 2 ) that any left noetherian ring ...
... endomorphism of M is an automorphism . ( 1 ) Show that any noetherian module M is hopfian . ( 2 ) Show that the left regular module RR is hopfian iff R is Dedekind- finite . ( 3 ) Deduce from ( 1 ) , ( 2 ) that any left noetherian ring ...
Page 12
... endomorphisms on the left . Let ɛ ; : M → nM be the jth inclusion , and π ; : nM → M be the ith projection . For any endomorphism F : nM → nM , let fij be the composition ¡ Fɛ ; € E. Define a map a : EndR ( nM ) → M , ( E ) by a ( F ) ...
... endomorphisms on the left . Let ɛ ; : M → nM be the jth inclusion , and π ; : nM → M be the ith projection . For any endomorphism F : nM → nM , let fij be the composition ¡ Fɛ ; € E. Define a map a : EndR ( nM ) → M , ( E ) by a ( F ) ...
Page 262
... endomorphism rings of modules . Let R = End ( MA ) , where M is a right module over some ring A. Armendariz , Fisher and Snider ( Comm . Algebra 6 ( 1978 ) , 659-672 ) have shown that R is strongly π - regular iff M has the " Fitting ...
... endomorphism rings of modules . Let R = End ( MA ) , where M is a right module over some ring A. Armendariz , Fisher and Snider ( Comm . Algebra 6 ( 1978 ) , 659-672 ) have shown that R is strongly π - regular iff M has the " Fitting ...
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