( φ , ϕ ) -derivations on semiprime rings and Banach algebras

Bilal Ahmad Wani

Communications in Mathematics (2021)

  • Volume: 29, Issue: 3, page 371-383
  • ISSN: 1804-1388

Abstract

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Let ℛ be a semiprime ring with unity e and φ , ϕ be automorphisms of ℛ . In this paper it is shown that if ℛ satisfies 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) for all x ∈ ℛ and some fixed integer n ≥ 2 , then 𝒟 is an ( φ , ϕ )-derivation. Moreover, this result makes it possible to prove that if ℛ admits an additive mappings 𝒟 , 𝒢 : ℛ → ℛ satisfying the relations 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒢 ( x ) + 𝒢 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒢 ( x n - 1 ) , 2 𝒢 ( x n ) = 𝒢 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) , for all x ∈ ℛ and some fixed integer n ≥ 2 , then 𝒟 and 𝒢 are ( φ , ϕ )derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.

How to cite

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Wani, Bilal Ahmad. "$(\phi , \varphi )$-derivations on semiprime rings and Banach algebras." Communications in Mathematics 29.3 (2021): 371-383. <http://eudml.org/doc/297499>.

@article{Wani2021,
abstract = {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 $n\ge 2$, 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 $n\ge 2$, then $\mathcal \{D\} $ and $\mathcal \{G\} $ are ($\phi $, $\varphi $)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.},
author = {Wani, Bilal Ahmad},
journal = {Communications in Mathematics},
keywords = {Prime ring; semiprime ring; Banach algebra; Jordan derivation; $(\phi , \varphi )$-derivation},
language = {eng},
number = {3},
pages = {371-383},
publisher = {University of Ostrava},
title = {$(\phi , \varphi )$-derivations on semiprime rings and Banach algebras},
url = {http://eudml.org/doc/297499},
volume = {29},
year = {2021},
}

TY - JOUR
AU - Wani, Bilal Ahmad
TI - $(\phi , \varphi )$-derivations on semiprime rings and Banach algebras
JO - Communications in Mathematics
PY - 2021
PB - University of Ostrava
VL - 29
IS - 3
SP - 371
EP - 383
AB - 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 $n\ge 2$, 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 $n\ge 2$, then $\mathcal {D} $ and $\mathcal {G} $ are ($\phi $, $\varphi $)derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras.
LA - eng
KW - Prime ring; semiprime ring; Banach algebra; Jordan derivation; $(\phi , \varphi )$-derivation
UR - http://eudml.org/doc/297499
ER -

References

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  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, (2003) MR1964528
  2. Ashraf, M., Rehman, N., 10.1515/dema-2004-0303, Demonstratio Mathematica, 37, 2, 2004, 275-284, (2004) MR2093532DOI10.1515/dema-2004-0303
  3. Bonsall, F.F., Duncan, J., Complete Normed Algebras, 1973, Springer-Verlag, New York, (1973) Zbl0271.46039
  4. Brešar, M., 10.1090/S0002-9939-1988-0929422-1, Proceedings of the American Mathematical Society, 104, 4, 1988, 1003-1006, (1988) DOI10.1090/S0002-9939-1988-0929422-1
  5. Brešar, M., 10.1016/0021-8693(89)90285-8, Journal of Algebra, 127, 1, 1989, 218-228, Elsevier, (1989) DOI10.1016/0021-8693(89)90285-8
  6. Brešar, M., Vukman, J., 10.1017/S0004972700026927, Bulletin of the Australian Mathematical Society, 37, 3, 1988, 321-322, Cambridge University Press, (1988) DOI10.1017/S0004972700026927
  7. Brešar, M., Vukman, J., Jordan ( θ , φ )-derivations, Glasnik Matematicki, 16, 1991, 13-17, (1991) 
  8. Cusack, J.M., 10.1090/S0002-9939-1975-0399182-5, Proceedings of the American Mathematical Society, 53, 2, 1975, 321-324, (1975) DOI10.1090/S0002-9939-1975-0399182-5
  9. Fošner, A., Vukman, J., 10.1007/s00605-009-0154-7, Monatshefte für Mathematik, 162, 2, 2011, 157-165, Springer, (2011) MR2769884DOI10.1007/s00605-009-0154-7
  10. Herstein, I.N., Jordan derivations of prime rings, Proceedings of the American Mathematical Society, 8, 6, 1957, 1104-1110, JSTOR, (1957) 
  11. Liu, C. K., Shiue, W. K., Generalized Jordan triple ( θ , φ ) -derivations on semiprime rings, Taiwanese Journal of Mathematics, 11, 5, 2007, 1397-1406, The Mathematical Society of the Republic of China, (2007) MR2368657
  12. Rehman, N., Širovnik, N., Bano, T., 10.1007/s00009-016-0823-4, Mediterranean Journal of Mathematics, 14, 1, 2017, 1-10, Springer, (2017) MR3589928DOI10.1007/s00009-016-0823-4
  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, (2016) MR3456519
  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., (2014) MR3185789
  15. Širovnik, N., Vukman, J., On certain functional equation in semiprime rings, Algebra Colloquium, 23, 1, 2016, 65-70, World Scientific, (2016) MR3439878
  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, (2012) 
  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, (2011) 
  18. Vukman, J., 10.1155/IJMMS.2005.1031, International Journal of Mathematics and Mathematical Sciences, 2005, 7, 2005, 1031-1038, Hindawi, (2005) MR2170502DOI10.1155/IJMMS.2005.1031
  19. Vukman, J., 10.11650/twjm/1500404650, Taiwanese Journal of Mathematics, 11, 1, 2007, 255-265, The Mathematical Society of the Republic of China, (2007) MR2304020DOI10.11650/twjm/1500404650
  20. Vukman, J., Kosi-Ulbl, I., 10.1155/IJMMS.2005.3347, International Journal of Mathematics and Mathematical Sciences, 2005, 20, 2005, 3347-3350, Hindawi, (2005) MR2208058DOI10.1155/IJMMS.2005.3347

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