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An identity with generalized derivations on Lie ideals, right ideals and Banach algebras

Vincenzo de Filippis; Giovanni Scudo; Mohammad S. Tammam El-Sayiad — 2012

Czechoslovak Mathematical Journal

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 $α \in C$ such that $F (x) = α x$ for all $x \in R$ ; (2) $R$ satisfies the standard identity $s_{4}$ and there exist $a \in U$ and $α \in C$ such that $F (x) = a x + x a + α x$ 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...

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