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Keywords:
$(\theta ,\varphi )$-derivation; Jordan $(\theta ,\varphi )$-derivation; Jordan triple $(\theta ,\varphi )$-derivation
$(\theta ,\varphi )$-derivation; Jordan $(\theta ,\varphi )$-derivation; Jordan triple $(\theta ,\varphi )$-derivation
Summary:
Let $R$ be a 2-torsion free prime ring and let $\theta ,\varphi $ be endomorphisms of $R$. We prove that if $R$ is commutative, then every Jordan triple $(\theta ,\varphi )$-derivation of $R$ is a \hbox {$(\theta ,\varphi )$-derivation} and if $R$ is noncommutative, then every Jordan triple $(\theta ,\varphi )$-derivation of $R$ with either $\theta $ or $\varphi $ an epimorphism, is a $(\theta ,\varphi )$-derivation. As an application, we characterize Jordan triple semiderivations of prime rings. Our theorems naturally generalize the result for Jordan $(\theta ,\varphi )$-derivations obtained by M. Brešar, J. Vukman (1991) and the result for Jordan semiderivations obtained by V. De Filippis, A. Mamouni, L. Oukhtite (2015).
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