Document Type: Research Paper

Author

Department of Mathematics, Kashmar Higher Education Institute, Kashmar, Iran

Abstract

The main purpose of this article is to offer some characterizations of $\delta$-double derivations on rings and algebras. To reach this goal, we prove the following theorem:
Let $n > 1$ be an integer and let $\mathcal{R}$ be an $n!$-torsion free ring with the identity element $1$. Suppose that there exist two additive mappings $d,\delta:R\to R$ such that $$d(x^n) =\Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+\Sigma^{n-2}_{k=0} \Sigma^{n-2-k}_{i=0} x^k\delta(x)x^i\delta(x)x^{n-2-k-i}$$ is ful filled for all $x\in \mathcal{R}$. If $\delta(1) = 0$, then $d$ is a Jordan $\delta$-double derivation. In particular, if $\mathcal{R}$ is a semiprime algebra and further, $\delta^2(x^2) = \delta^2(x)x + x\delta^2(x) + 2(\delta(x))^2$ holds for all $x\in \mathcal{R}$, then $d-\frac{1}{2}\delta^2$ is an ordinary derivation on $\mathcal{R}$.

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