Characterization of $delta$-double derivations on rings and algebras

author

  • A. Hosseini 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 $d,delta:Rto R$ such that $$d(x^n) =Sigma^n_{j=1} x^{n-j}d(x)x^{j-1}+Sigma^{n-2}_{k=0} Sigma^{n-2-k}_{i=0} x^kdelta(x)x^idelta(x)x^{n-2-k-i}$$ is fulfilled for all $xin mathcal{R}$. If $delta(1) = 0$, then $d$ is a Jordan $delta$-double derivation. In particular, if $mathcal{R}$ is a semiprime algebra and further, $delta^2(x^2) = delta^2(x)x + xdelta^2(x) + 2(delta(x))^2$ holds for all $xin mathcal{R}$, then $d-frac{1}{2}delta^2$ is an ordinary derivation on $mathcal{R}$.

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Journal title

volume 06  issue 01

pages  55- 65

publication date 2017-03-01

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