Characterization of $\delta$-double derivations on rings and algebras
محورهای موضوعی : History and biography
1 - Department of Mathematics, Kashmar Higher Education Institute, Kashmar, Iran
کلید واژه: $n$-torsion free semiprime ring, $delta$-Double derivation, Jordan $delta$-double derivation,
چکیده مقاله :
The main purpose of this article is to offer some characterizations of $\delta$-doublederivations 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 fulfilled 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 $\mathcal{R}$.
[1] D. Bridges, J. Bergen, On the derivation of $x_n$ in a ring, Proc. Amer. Math. Soc. 90 (1984), 25-29.
[2] M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 140 (4) (1988), 1003-1006.
[3] M. Bresar, J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37 (1988), 321-322.
[4] M. Bresar, Characterizations of derivations on some normed algebras with involution, Journal of Algebra. 152 (1992), 454-462.
[5] J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 1104-1110.
[6] H. G. Dales, P. Aiena, J. Eschmeier, K. Laursen, G. A. Willis, Introduction to Banach Algebras, Operators and Harmonic Analysis. Cambridge University Press, 2003.
[7] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110.
[8] A. Hosseini, A characterization of weak $n_m$-Jordan (alpha,beta)-derivations by generalized centralizers, Rend. Circ. Mat. Palermo. 64 (2015), 221-227.
[9] M. Mirzavaziri and E. Omidvar Tehrani, delta-double derivations on C*-algebras, Bull. Iranian. Math .Soc. 35 (2009), 147-154.
[10] J. Vukman, On left Jordan derivations of rings and Banach algebras, Aequ. Math. 75 (2008), 260-266.
[11] J. Vukman, A note on generalized derivations of semiprime rings, Taiwanese Journal of Mathematics. 11 (2) (2007), 367-370.
[12] J. Vukman, I. Kosi-Ulbl, On derivations in rings with involution, Int. Math. J. 6 (2005), 81-91.
[13] J. Vukman, I. Kosi-Ulbl, On some equations related to derivations in rings, Int. J. Math. Math. Sci. 17 (2005), 2703-2710.
