Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 39
... maximal left ideals ( see Exercise 5 ) , some of the above verifications can be simplified . However , modularity is ... ideal ICR is said to be modular ( or regular2 ) if there exists e Є R which serves as a " right identity mod I ” ; i.e. ...
... maximal left ideals ( see Exercise 5 ) , some of the above verifications can be simplified . However , modularity is ... ideal ICR is said to be modular ( or regular2 ) if there exists e Є R which serves as a " right identity mod I ” ; i.e. ...
Page 42
... maximal ideals of R , show that rad RC rad'R , and give an example to show that this may be a strict inclusion . ( rad ' R is called the Brown - McCoy radical of R. ) Solution . Let V = R / m , where m is a maximal left ideal ... ideal of R ...
... maximal ideals of R , show that rad RC rad'R , and give an example to show that this may be a strict inclusion . ( rad ' R is called the Brown - McCoy radical of R. ) Solution . Let V = R / m , where m is a maximal left ideal ... ideal of R ...
Page 51
... maximal left ideal , we must show that yr Є m . Assume otherwise ; then Rr + m = R. Consider the left R - module homomorphism : R → R / m defined by ( x ) = xr . Since Rr + m = R , is onto . This implies that ker ( ) is a maximal left ...
... maximal left ideal , we must show that yr Є m . Assume otherwise ; then Rr + m = R. Consider the left R - module homomorphism : R → R / m defined by ( x ) = xr . Since Rr + m = R , is onto . This implies that ker ( ) is a maximal left ...
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