Abstract
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.
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RAZA, M.A., REHMAN, N.U. On prime and semiprime rings with generalized derivations and non-commutative Banach algebras. Proc Math Sci 126, 389–398 (2016). https://doi.org/10.1007/s12044-016-0287-2
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DOI: https://doi.org/10.1007/s12044-016-0287-2