Exercises in Classical Ring TheorySpringer-Verlag, 1995 - 287 pages |
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Page 5
... identity e . Since BB1 = 0 for ij , it follows that each e ; is central . We finish easily by showing that defined ... identity ) of order p2 is commutative . ( b ) Show that there exists a noncommutative ring without identity of order ...
... identity e . Since BB1 = 0 for ij , it follows that each e ; is central . We finish easily by showing that defined ... identity ) of order p2 is commutative . ( b ) Show that there exists a noncommutative ring without identity of order ...
Page 6
... identity of order p3 . ( b ) For any nonzero ring k , R a b 0 a , 0 = { ( ° ) : ab € k } is a right ideal of M2 ( k ) . Therefore , ( R , + , × ) satisfies all the axioms of a ring , except perhaps the identity axiom . An easy ...
... identity of order p3 . ( b ) For any nonzero ring k , R a b 0 a , 0 = { ( ° ) : ab € k } is a right ideal of M2 ( k ) . Therefore , ( R , + , × ) satisfies all the axioms of a ring , except perhaps the identity axiom . An easy ...
Page 36
... identity . This is not essential for the problems in the later sections , but it is nice to know how this the- ory ... identity element . In particular , the theory of the Jacobson radical was developed in that text for rings with an ...
... identity . This is not essential for the problems in the later sections , but it is nice to know how this the- ory ... identity element . In particular , the theory of the Jacobson radical was developed in that text for rings with an ...
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