Document Type: Research Paper

Authors

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

Abstract

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 $\mathcal{R}$.

Keywords

Main Subjects

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