Characterization of $(\delta, \varepsilon)$-double derivation on rings and algebras
محورهای موضوعی : Operator theory
1 - Department of Mathematics, Mashhad Branch, Islamic Azad University-Mashhad, Iran
2 - Department of Mathematics, Ferdowsi University of Mashhad and Center of Excellence in Analysis on Algebraic Structures (CEAAS) Ferdowsi University, Mashhad, Iran
کلید واژه: derivation, Jordan derivation, (&delta, , &lrm, &epsilon, )-double derivation, &lrm, n-torsion free semiprime ring,
چکیده مقاله :
This paper is an attempt to prove the following result:Let $n>1$ be an integer and let $\mathcal{R}$ be a $n!$-torsion-free ring with the identity element. Suppose that $d, \delta, \varepsilon$ are additive mappings satisfying\begin{equation}d(x^n) = \sum^{n}_{j=1}x^{n-j}d(x)x^{j-1}+\sum^{n-1}_{j=1}\sum^{j}_{i=1}x^{n-1-j}\Big(\delta(x)x^{j-i}\varepsilon(x)+\varepsilon(x)x^{j-i}\delta(x)\Big)x^{i-1}\quad\end{equation}for all $x \in \mathcal{R}$. If $\delta(e) = \varepsilon(e) = 0$, then $d$ is a Jordan $(\delta, \varepsilon)$-double derivation.In particular, if $\mathcal{R}$ is a semiprime algebra and further, $\delta(x) \varepsilon(x) + \varepsilon(x) \delta(x) = \frac{1}{2}\Big[(\delta \varepsilon + \varepsilon \delta)(x^2) - (\delta \varepsilon(x) + \varepsilon \delta(x))x - x (\delta \varepsilon(x) + \varepsilon \delta(x))\Big]$ holds for all $x \in \mathcal{R}$, then $d - \frac{\delta \varepsilon + \varepsilon \delta}{2}$ is a derivation on $\mathcal{R}$.
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