Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 27
... infinite - dimensional algebra over a field k . Show that any nonzero left R - module V is also infinite - dimensional over k . Solution . The left action of R on V leads to a k - algebra homomorphism ❤ : R → End ( V ) . Since V 0 and ...
... infinite - dimensional algebra over a field k . Show that any nonzero left R - module V is also infinite - dimensional over k . Solution . The left action of R on V leads to a k - algebra homomorphism ❤ : R → End ( V ) . Since V 0 and ...
Page 83
... infinite order in ZG , with inverse v = = 1− x − x1 . Then show that U ( ZG ) = ( u ) × ( ± G ) ≈ Z ® Z2 → Z5 . - For the more computationally inclined reader , show that a = 2x1 — x3 3x2 - x + 2 is a unit of infinite order in ZG ...
... infinite order in ZG , with inverse v = = 1− x − x1 . Then show that U ( ZG ) = ( u ) × ( ± G ) ≈ Z ® Z2 → Z5 . - For the more computationally inclined reader , show that a = 2x1 — x3 3x2 - x + 2 is a unit of infinite order in ZG ...
Page 239
... infinite set . We can write A = BUC where B , C are disjoint infinite sets . Then A is also the symmetric difference of B and C , so A + is the sum of the two nonzero orthogonal idempotents B + and C + in R. Therefore , A + cannot be a ...
... infinite set . We can write A = BUC where B , C are disjoint infinite sets . Then A is also the symmetric difference of B and C , so A + is the sum of the two nonzero orthogonal idempotents B + and C + in R. Therefore , A + cannot be a ...
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