Characterization of $(\delta‎, ‎\varepsilon)$-double derivation on rings ‎and ‎algebras

Journal of Linear and Topological Algebra

مقاله 1، دوره 06، شماره 03، آذر 2017، صفحه 191-198   

نوع مقاله: Research Paper

نویسندگان

Z. Jokar^{* 1}؛ A. Niknam²

¹Department of Mathematics, Mashhad Branch, Islamic Azad University-Mashhad, Iran

²Department of Mathematics, Ferdowsi University of Mashhad and Center of Excellence in Analysis on Algebraic Structures (CEAAS) Ferdowsi University, Mashhad, Iran

چکیده

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 $\mathcal{R}$.

کلیدواژه‌ها

derivation؛ Jordan derivation؛ (δ,‎ε)-double derivation؛ ‎n-torsion free semiprime ring

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