Derivations with Power Values on Lie Ideals in Rings and Banach Algebras
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Let $R$ be a $2$-torsion free prime ring with center $Z$, $U$ be the Utumi quotient ring, $Q$ be the Martindale quotient ring of $R$, $d$ be a derivation of $R$ and $L$ be a Lie ideal of $R$. If $d(uv)^n=d(u)^md(v)^l$ or $d(uv)^n=d(v)^ld(u)^m$ for all $u,v\in L$, where $m,n,l$ are fixed positive integers, then $L \subseteq Z$. We also examine the case when $R$ is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non-commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.
Abstract
Let $R$ be a $2$-torsion free prime ring with center $Z$, $U$ be the Utumi quotient ring, $Q$ be the Martindale quotient ring of $R$, $d$ be a derivation of $R$ and $L$ be a Lie ideal of $R$. If $d(uv)^n=d(u)^md(v)^l$ or $d(uv)^n=d(v)^ld(u)^m$ for all $u,v\in L$, where $m,n,l$ are fixed positive integers, then $L \subseteq Z$. We also examine the case when $R$ is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non-commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.
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