Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 41
... radical : if this is the case , R is said to be a radical ring . Show that R is a radical ring iff it has no simple left ( resp . right ) modules . Solution . Let m 2 rad R be a typical modular maximal left ideal . Clearly m / rad R is ...
... radical : if this is the case , R is said to be a radical ring . Show that R is a radical ring iff it has no simple left ( resp . right ) modules . Solution . Let m 2 rad R be a typical modular maximal left ideal . Clearly m / rad R is ...
Page 42
... radical of R. ) Solution . Let V = R / m , where m is a maximal left ideal of R containing I. Then IV = 0 , so I ... radical rad ' R may be defined as the intersection of all modular maximal ideals of R ( where " modular " is now ...
... radical of R. ) Solution . Let V = R / m , where m is a maximal left ideal of R containing I. Then IV = 0 , so I ... radical rad ' R may be defined as the intersection of all modular maximal ideals of R ( where " modular " is now ...
Page 103
... radical ) . The lower nilradical ( or Baer - McCoy radical ) , denoted by Nil✶R , is simply √ ( 0 ) . This is the intersection of all the prime ideals of R , and , for this reason , it is also known as the prime radical . The upper ...
... radical ) . The lower nilradical ( or Baer - McCoy radical ) , denoted by Nil✶R , is simply √ ( 0 ) . This is the intersection of all the prime ideals of R , and , for this reason , it is also known as the prime radical . The upper ...
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