Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u ε L, then one of the following holds: there exists α ε C such that F(x) = αx for all x ε R; R satisfies the standard identity s 4 and there exist a ε U and α ε C such that F(x) = ax + xa + αx for all x ε R. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras. © 2012 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.
An identity with generalized derivations on lie ideals, right ideals and Banach algebras
DE FILIPPIS, Vincenzo;SCUDO, GIOVANNI;
2012-01-01
Abstract
Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u ε L, then one of the following holds: there exists α ε C such that F(x) = αx for all x ε R; R satisfies the standard identity s 4 and there exist a ε U and α ε C such that F(x) = ax + xa + αx for all x ε R. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras. © 2012 Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.Pubblicazioni consigliate
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