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

Subject Areas : Operator theory

Z. Jokar ¹ , A. Niknam ²

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

Received: 2017-08-09 Accepted : 2017-11-12 Published : 2017-12-01

Keywords: derivation, Jordan derivation, (&delta, , &lrm, &epsilon, )-double derivation, &lrm, n-torsion free semiprime ring,

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 $\mathcal{R}$.

References:

[1] M. Brear, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 140 (4) (1988), 1003-1006.

[2] M. Brear, J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37 (1988), 321-322.

[3] M. Brear, Characterizations of derivations on some normed algebras with involution, Journal of  Algebra. 152 (1992), 454-462.

[4] D. Bridges, J. Bergen, On the derivation of xn in a ring, Proc. Amer. Math. Soc. 90 (1984), 25-29.

[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, 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 δ-double derivations on rings and algebras, Journal of Linear and Topological Algebra. 06 (01) (2017), 55-65.

[9] M. Mirzavaziri, E. Omidvar Tehrani, δ-Double derivations on c*-algebras, Bulletin of the  Iranian Mathematical Society. 35 (1) (2009), 147-154.

[10] J. Vukman, I. Kosi-Ulbl, On derivations in rings with involution, Int. Math. J. 6 (2005), 81-91.

[11] J. Vukman, I. Kosi-Ulbl, On some equations related to derivations in rings, Int. J. Math. Math. Sci. 17 (2005), 2703-2710.

[12] J. Vukman, I. Kosi-Ulbl, A note on derivation in semiprime rings, Int. J. Math. Math. Sci. 17 (2005), 3347-3350.

[13] J. Vukman, A note on generalized derivations of semiprime rings, Taiwanese Journal of Mathematics. 11 (2) (2007), 367-370.