# Article

 Title: An identity with generalized derivations on Lie ideals, right ideals and Banach algebras (English) Author: de Filippis, Vincenzo Author: Scudo, Giovanni Author: Tammam El-Sayiad, Mohammad S. Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 62 Issue: 2 Year: 2012 Pages: 453-468 Summary lang: English . Category: math . Summary: 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 $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F(x)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity $s_4$ and there exist $a\in U$ and $\alpha \in C$ such that $F(x)=ax+xa+\alpha x$ for all $x\in R$. 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. (English) Keyword: prime rings Keyword: differential identities Keyword: generalized derivations Keyword: Banach algebra MSC: 16N60 MSC: 16W25 MSC: 47B47 MSC: 47B48 idZBL: Zbl 1249.16045 idMR: MR2990186 DOI: 10.1007/s10587-012-0039-0 . Date available: 2012-06-08T09:45:01Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/142838 . Reference: [1] Beidar, K. I.: Rings with generalized identities. III.Mosc. Univ. Math. Bull. 33 (1978), 53-58. Zbl 0407.16002, MR 0510966 Reference: [2] Beidar, K. I., III, W. S. Martindale, Mikhalev, A. V.: Rings with Generalized Identities.Pure and Applied Mathematics, Marcel Dekker. 196. New York (1996). MR 1368853 Reference: [3] Brešar, M., Mathieu, M.: Derivations mapping into the radical III.J. Funct. Anal. 133 (1995), 21-29. 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