Article

Full entry | PDF   (0.3 MB)
Keywords:
Prime ring; semiprime ring; Banach algebra; Jordan derivation; $(\phi, \varphi )$-derivation
Summary:
Let $\mathcal{R}$ be a semiprime ring with unity $e$ and $\phi$, $\varphi$ be automorphisms of $\mathcal{R}$. In this paper it is shown that if $\mathcal{R}$ satisfies $$2\mathcal{D} (x^n) = \mathcal{D} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{D} (x)+\mathcal{D} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{D} (x^{n-1})$$ for all $x\in \mathcal{R}$ and some fixed integer $n\geq 2$, then $\mathcal{D}$ is an ($\phi$, $\varphi$)-derivation. Moreover, this result makes it possible to prove that if $\mathcal { R}$ admits an additive mappings $\mathcal{D} ,\mathcal{G} \colon \mathcal{R} \rightarrow \mathcal{R}$ satisfying the relations \begin {gather*}\nonumber 2\mathcal{D} (x^n) = \mathcal{D} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{G} (x)+\mathcal{G} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{G} (x^{n-1})\,, \\ 2\mathcal{G} (x^n) = \mathcal{G} (x^{n-1})\phi (x) + \varphi (x^{n-1})\mathcal{D} (x)+\mathcal{D} (x)\phi (x^{n-1}) + \varphi (x)\mathcal{D} (x^{n-1})\,, \end {gather*} for all $x\in \mathcal{R}$ and some fixed integer $n\geq 2$, then $\mathcal{D}$ and $\mathcal{G}$ are ($\phi$, $\varphi$)\HH derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.
References:
[1] Ashraf, M., Rehman, N., Ali, S.: On Lie ideals and Jordan generalized derivations of prime rings. Indian Journal of Pure & Applied Mathematics, 34, 2, 2003, 291-294, King Abdulaziz University, MR 1964528
[2] Ashraf, M., Rehman, N.: On Jordan ideals and Jordan derivations of a prime rings. Demonstratio Mathematica, 37, 2, 2004, 275-284, DOI 10.1515/dema-2004-0303 | MR 2093532
[3] Bonsall, F.F., Duncan, J.: Complete Normed Algebras. 1973, Springer-Verlag, New York, Zbl 0271.46039
[4] Brešar, M.: Jordan derivations on semiprime rings. Proceedings of the American Mathematical Society, 104, 4, 1988, 1003-1006, DOI 10.1090/S0002-9939-1988-0929422-1
[5] Brešar, M.: Jordan mappings of semiprime rings. Journal of Algebra, 127, 1, 1989, 218-228, Elsevier, DOI 10.1016/0021-8693(89)90285-8
[6] Brešar, M., Vukman, J.: Jordan derivations on prime rings. Bulletin of the Australian Mathematical Society, 37, 3, 1988, 321-322, Cambridge University Press, DOI 10.1017/S0004972700026927
[7] Brešar, M., Vukman, J.: Jordan ($\theta$, $\phi$)-derivations. Glasnik Matematicki, 16, 1991, 13-17,
[8] Cusack, J.M.: Jordan derivations on rings. Proceedings of the American Mathematical Society, 53, 2, 1975, 321-324, DOI 10.1090/S0002-9939-1975-0399182-5
[9] Fošner, A., Vukman, J.: On certain functional equations related to Jordan triple $(\theta ,\phi )$-derivations on semiprime rings. Monatshefte für Mathematik, 162, 2, 2011, 157-165, Springer, DOI 10.1007/s00605-009-0154-7 | MR 2769884
[10] Herstein, I.N.: Jordan derivations of prime rings. Proceedings of the American Mathematical Society, 8, 6, 1957, 1104-1110, JSTOR,
[11] Liu, C. K., Shiue, W. K.: Generalized Jordan triple $(\theta ,\phi )$-derivations on semiprime rings. Taiwanese Journal of Mathematics, 11, 5, 2007, 1397-1406, The Mathematical Society of the Republic of China, MR 2368657
[12] Rehman, N., Širovnik, N., Bano, T.: On certain functional equations on standard operator algebras. Mediterranean Journal of Mathematics, 14, 1, 2017, 1-10, Springer, DOI 10.1007/s00009-016-0823-4 | MR 3589928
[13] Rehman, N., Bano, T.: A result on functional equations in semiprime rings and standard operator algebras. Acta Mathematica Universitatis Comenianae, 85, 1, 2016, 21-28, MR 3456519
[14] Širovnik, N.: On certain functional equation in semiprime rings and standard operator algebras. Cubo (Temuco), 16, 1, 2014, 73-80, Universidad de La Frontera. Departamento de Matemática y Estadística., MR 3185789
[15] Širovnik, N., Vukman, J.: On certain functional equation in semiprime rings. Algebra Colloquium, 23, 1, 2016, 65-70, World Scientific, MR 3439878
[16] Širovnik, N.: On functional equations related to derivations in semiprime rings and standard operator algebras. Glasnik Matematički, 47, 1, 2012, 95-104, Hrvatsko matematičko društvo i PMF-Matematički odjel, Sveučilišta u Zagrebu,
[17] Vukman, J.: Some remarks on derivations in semiprime rings and standard operator algebras. Glasnik Matematički, 46, 1, 2011, 43-48, Hrvatsko matematičko društvo i PMF-Matematički odjel, Sveučilišta u Zagrebu,
[18] Vukman, J.: Identities with derivations and automorphisms on semiprime rings. International Journal of Mathematics and Mathematical Sciences, 2005, 7, 2005, 1031-1038, Hindawi, DOI 10.1155/IJMMS.2005.1031 | MR 2170502
[19] Vukman, J.: Identities related to derivations and centralizers on standard operator algebras. Taiwanese Journal of Mathematics, 11, 1, 2007, 255-265, The Mathematical Society of the Republic of China, DOI 10.11650/twjm/1500404650 | MR 2304020
[20] Vukman, J., Kosi-Ulbl, I.: A note on derivations in semiprime rings. International Journal of Mathematics and Mathematical Sciences, 2005, 20, 2005, 3347-3350, Hindawi, DOI 10.1155/IJMMS.2005.3347 | MR 2208058

Partner of