Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 16
... module is isomorphic to R / m for some maximal left ideal m in R. A general module is " seldom " semisimple . For instance , over R = Z , M = Z / 4Z is not semisimple , since N = 2Z / 4Z is not a direct summand . Over the polynomial ring ...
... module is isomorphic to R / m for some maximal left ideal m in R. A general module is " seldom " semisimple . For instance , over R = Z , M = Z / 4Z is not semisimple , since N = 2Z / 4Z is not a direct summand . Over the polynomial ring ...
Page 17
... semisimple ring left semisimple ? Can any ring be embedded as a subring of a left semisimple ring ? ... Solution . The answer to the first question is " no " : for instance , Z is a subring of the semisimple ring Q , but Z is not a ...
... semisimple ring left semisimple ? Can any ring be embedded as a subring of a left semisimple ring ? ... Solution . The answer to the first question is " no " : for instance , Z is a subring of the semisimple ring Q , but Z is not a ...
Page 22
... ring is a field , and the center of a semisimple ring is a finite direct product of fields . Solution . Suppose R is a simple ring , and let 0 a € Z ( R ) . Then Ra is an ideal , so Ra = R. This implies that a € U ( R ) . But clearly a1 ...
... ring is a field , and the center of a semisimple ring is a finite direct product of fields . Solution . Suppose R is a simple ring , and let 0 a € Z ( R ) . Then Ra is an ideal , so Ra = R. This implies that a € U ( R ) . But clearly a1 ...
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