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An identity with generalized derivations on Lie ideals, right ideals and Banach algebras
Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 453-468
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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.
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.
DOI : 10.1007/s10587-012-0039-0
Classification : 16N60, 16W25, 47B47, 47B48
Keywords: prime rings; differential identities; generalized derivations; Banach algebra 
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     title = {An identity with generalized derivations on {Lie} ideals, right ideals and {Banach} algebras},
     journal = {Czechoslovak Mathematical Journal},
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de Filippis, Vincenzo; Scudo, Giovanni; Tammam El-Sayiad, Mohammad S. An identity with generalized derivations on Lie ideals, right ideals and Banach algebras. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 453-468. doi: 10.1007/s10587-012-0039-0

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