On prime and semiprime rings with generalized derivations and non-commutative Banach algebras
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Let R be a prime ring of characteristic different from 2 and m a fixed positive integer. If R admits a generalized derivation associated with a nonzero deviation d such that [F(x),d(y)] m =[x,y] for all x,y in some appropriate subset of R, then R is commutative. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer–Werner theorem.
KeywordsBanach algebras generalized derivations martindale ring of quotients prime and semiprime rings radical
2010 Mathematics Subject Classification46J10 16N20 16N60 16W25
The authors wish to thank the referee for his/her valuable comments and suggestions.
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