Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 36
... element . In particular , the theory of the Jacobson radical was developed in that text for rings with an identity . However , by doing things a little more carefully , the whole theory can be carried over to rings possibly without an ...
... element . In particular , the theory of the Jacobson radical was developed in that text for rings with an identity . However , by doing things a little more carefully , the whole theory can be carried over to rings possibly without an ...
Page 65
... element b − 1 Є A. Then 1 € supp ( ba ) , and ba ( ba ) * implies that | supp ( a ) | = | supp ( ba ) | is odd , since A has no element of order 2. But if supp ( a ) misses some element c ̄1 € A , then 1 & supp ( ca ) , and ca = ( ca ) ...
... element b − 1 Є A. Then 1 € supp ( ba ) , and ba ( ba ) * implies that | supp ( a ) | = | supp ( ba ) | is odd , since A has no element of order 2. But if supp ( a ) misses some element c ̄1 € A , then 1 & supp ( ca ) , and ca = ( ca ) ...
Page 168
... elements of the above set which are < N. Therefore , we can again " rearrange " the elements in ( 2 ) above in a nondecreasing sequence ( which tends to ∞ ) . It follows that the RHS of ( 1 ) is an element of A1 , so A1 is a ( proper ) ...
... elements of the above set which are < N. Therefore , we can again " rearrange " the elements in ( 2 ) above in a nondecreasing sequence ( which tends to ∞ ) . It follows that the RHS of ( 1 ) is an element of A1 , so A1 is a ( proper ) ...
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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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