Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 8
... element of A is algebraic over k . ( a ) Show that A is Dedekind - finite . ( b ) Show that a left 0 - divisor of A is also a right 0 - divisor . ( c ) Show that a nonzero element of A is a unit iff it is not a 0 - divisor . ( d ) Let B ...
... element of A is algebraic over k . ( a ) Show that A is Dedekind - finite . ( b ) Show that a left 0 - divisor of A is also a right 0 - divisor . ( c ) Show that a nonzero element of A is a unit iff it is not a 0 - divisor . ( d ) Let B ...
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 € 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 c1 A , then 1 supp ( ca ) , and ca = ( ca ) ...
... element b - 1 € 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 c1 A , then 1 supp ( ca ) , and ca = ( ca ) ...
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