Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 14
... isomorphic " as rings with involutions , " with isomorphism defined by a . Z nZ ) Show that RS , and that these are rings with involutions . Ex . 1.23 . For a fixed n ≥ 1 , let R = ( ( Z and S = Ꮓ Ꮓ ( Z Ꮓ nZ Z M2 ( Z ) defined by ...
... isomorphic " as rings with involutions , " with isomorphism defined by a . Z nZ ) Show that RS , and that these are rings with involutions . Ex . 1.23 . For a fixed n ≥ 1 , let R = ( ( Z and S = Ꮓ Ꮓ ( Z Ꮓ nZ Z M2 ( Z ) defined by ...
Page 46
... isomorphism from N to N ' . Write a = aua , where u € U ( R ) . As in Exercise 14A , ( * ) M = ker ( a ) im ( ua ) = K → u ( N ' ) . Since u defines an isomorphism from N ' to u ( N ' ) , it induces an isomor- phism from M / N ' to M ...
... isomorphism from N to N ' . Write a = aua , where u € U ( R ) . As in Exercise 14A , ( * ) M = ker ( a ) im ( ua ) = K → u ( N ' ) . Since u defines an isomorphism from N ' to u ( N ' ) , it induces an isomor- phism from M / N ' to M ...
Page 74
... isomorphism of K - vector spaces . Replacing the hypothesis dimдM < ∞ by " M is a finitely presented R - module , " give a basis - free proof for the fact that is a K - isomorphism . Solution . First consider the special case when M≈R ...
... isomorphism of K - vector spaces . Replacing the hypothesis dimдM < ∞ by " M is a finitely presented R - module , " give a basis - free proof for the fact that is a K - isomorphism . Solution . First consider the special case when M≈R ...
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