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Characterization of multiplicative Jordan n-higher derivations on unital rings

  • Ab Hamid Kawa ORCID logo , S. N. Hasan , Shakir Ali ORCID logo EMAIL logo und Bilal Ahmad Wani ORCID logo
Veröffentlicht/Copyright: 14. November 2024
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Georgian Mathematical Journal
Aus der Zeitschrift Georgian Mathematical Journal Band 32 Heft 3

Abstract

Let m ≥ 1 , n ≥ 3 be fixed integers, ℛ be a unital ring with a nontrivial idempotent, and let ζ m : ℛ → ℛ be a multiplicative Jordan n-higher derivation ( n ≥ 3 ) . In this paper, we prove that if c ⁢ h ⁢ a ⁢ r ⁢ ( ℛ ) ≠ 2 ⁢ 𝑎𝑛𝑑 ≠ n - 1 , then ζ m is an additive Jordan n-higher derivation, more precisely, ζ m ⁢ ( a ) = D m ⁢ ( a ) + π m ⁢ ( a ) for all a ∈ ℛ , where D m : ℛ → ℛ is an additive higher derivation and π m : ℛ → ℛ is an additive singular Jordan higher derivation.

Keywords: Derivation; Jordan derivation; unital ring
MSC 2020: 16W10; 47B47

Acknowledgements

The authors are very thankful to the anonymous referee for his/her valuable comments and suggestions, which have improved the manuscript immensely.

References

[1] M. Ashraf, B. A. Wani and F. Wei, Multiplicative * -Lie triple higher derivations of standard operator algebras, Quaest. Math. 42 (2019), no. 7, 857–884. 10.2989/16073606.2018.1502213Suche in Google Scholar

[2] D. Benkovič and N. Širovnik, Jordan derivations of unital algebras with idempotents, Linear Algebra Appl. 437 (2012), no. 9, 2271–2284. 10.1016/j.laa.2012.06.009Suche in Google Scholar

[3] M. Brešar, Jordan mappings of semiprime rings, J. Algebra 127 (1989), no. 1, 218–228. 10.1016/0021-8693(89)90285-8Suche in Google Scholar

[4] M. Brešar, Jordan derivations revisited, Math. Proc. Cambridge Philos. Soc. 139 (2005), no. 3, 411–425. 10.1017/S0305004105008601Suche in Google Scholar

[5] M. Brešar and J. Vukman, Jordan derivations on prime rings, Bull. Aust. Math. Soc. 37 (1988), no. 3, 321–322. 10.1017/S0004972700026927Suche in Google Scholar

[6] M. Brešar and J. Vukman, Jordan ( Θ , ϕ ) -derivations, Glas. Mat. Ser. III 26(46) (1991), no. 1–2, 13–17. Suche in Google Scholar

[7] J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), no. 2, 321–324. 10.1090/S0002-9939-1975-0399182-5Suche in Google Scholar

[8] M. Ferrero and C. Haetinger, Higher derivations of semiprime rings, Comm. Algebra 30 (2002), no. 5, 2321–2333. 10.1081/AGB-120003471Suche in Google Scholar

[9] A. Fošner, F. Wei and Z. Xiao, Nonlinear Lie-type derivations of von Neumann algebras and related topics, Colloq. Math. 132 (2013), no. 1, 53–71. 10.4064/cm132-1-5Suche in Google Scholar

[10] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110. 10.1090/S0002-9939-1957-0095864-2Suche in Google Scholar

[11] A. H. Kawa, S. N. Hasan and B. A. Wani, Multiplicative Jordan n-higher derivations on alternative rings containing idempotents, Asian-Eur. J. Math. 16 (2023), no. 5, Article ID 2350081. 10.1142/S179355712350081XSuche in Google Scholar

[12] A. H. Kawa, S. N. Hasan and B. A. Wani, Multiplicative Jordan type higher derivations of unital rings with non trivial idempotents, Adv. Pure Appl. Math. 14 (2023), no. 1, 36–49. 10.21494/ISTE.OP.2023.0905Suche in Google Scholar

[13] T.-K. Lee and T. C. Quynh, Centralizers and Jordan triple derivations of semiprime rings, Comm. Algebra 47 (2019), no. 1, 236–251. 10.1080/00927872.2018.1472275Suche in Google Scholar

[14] C. Li and X. Fang, Lie triple and Jordan derivable mappings on nest algebras, Linear Multilinear Algebra 61 (2013), no. 5, 653–666. 10.1080/03081087.2012.703186Suche in Google Scholar

[15] J. Li and F. Lu, Additive Jordan derivations of reflexive algebras, J. Math. Anal. Appl. 329 (2007), no. 1, 102–111. 10.1016/j.jmaa.2006.06.019Suche in Google Scholar

[16] W. H. Lin, Nonlinear ∗ -Jordan-type derivations on von Neumann algebras, preprint (2018), https://arxiv.org/abs/1805.02037. Suche in Google Scholar

[17] W. H. Lin, Nonlinear * -Lie-type derivations on standard operator algebras, Acta Math. Hungar. 154 (2018), no. 2, 480–500. 10.1007/s10474-017-0783-6Suche in Google Scholar

[18] W.-H. Lin, Nonlinear ∗ -Lie-type derivations on von Neumann algebras, Acta Math. Hungar. 156 (2018), no. 1, 112–131. 10.1007/s10474-018-0803-1Suche in Google Scholar

[19] F. Lu, Jordan derivable maps of prime rings, Comm. Algebra 38 (2010), no. 12, 4430–4440. 10.1080/00927870903366884Suche in Google Scholar

[20] W. S. Martindale, III, When are multiplicative mappings additive?, Proc. Amer. Math. Soc. 21 (1969), 695–698. 10.2307/2036449Suche in Google Scholar

[21] C. R. Miers, Lie homomorphisms of operator algebras, Pacific J. Math. 38 (1971), 717–735. 10.2140/pjm.1971.38.717Suche in Google Scholar

[22] A. Nowicki, Inner derivations of higher orders, Tsukuba J. Math. 8 (1984), no. 2, 219–225. 10.21099/tkbjm/1496160039Suche in Google Scholar

[23] X. Qi, Z. Guo and T. Zhang, Characterizing Jordan n-derivations of unital rings containing idempotents, Bull. Iranian Math. Soc. 46 (2020), no. 6, 1639–1658. 10.1007/s41980-019-00348-7Suche in Google Scholar

[24] X. Qi and J. Hou, Lie higher derivations on nest algebras, Commun. Math. Res. 26 (2010), no. 2, 131–143. Suche in Google Scholar

[25] A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc. 24 (1970), 209–214. 10.1090/S0002-9939-1970-0250069-3Suche in Google Scholar

[26] A. Taghavi, H. Rohi and V. Darvish, Non-linear *-Jordan derivations on von Neumann algebras, Linear Multilinear Algebra 64 (2016), no. 3, 426–439. 10.1080/03081087.2015.1043855Suche in Google Scholar

[27] J. Vukman and I. Kosi-Ulbl, A note on derivations in semiprime rings, Int. J. Math. Math. Sci. (2005), no. 20, 3347–3350. 10.1155/IJMMS.2005.3347Suche in Google Scholar

[28] F. Wei and Z. Xiao, Higher derivations of triangular algebras and its generalizations, Linear Algebra Appl. 435 (2011), no. 5, 1034–1054. 10.1016/j.laa.2011.02.027Suche in Google Scholar

[29] Z. Xiao and F. Wei, Nonlinear Lie higher derivations on triangular algebras, Linear Multilinear Algebra 60 (2012), no. 8, 979–994. 10.1080/03081087.2011.639373Suche in Google Scholar

[30] F. Zhang, Nonlinear skew Jordan derivable maps on factor von Neumann algebras, Linear Multilinear Algebra 64 (2016), no. 10, 2090–2103. 10.1080/03081087.2016.1139035Suche in Google Scholar

[31] F. Zhang, X. Qi and J. Zhang, Nonlinear 8-Lie higher derivations on factor von Neumann algebras, Bull. Iranian Math. Soc. 42 (2016), no. 3, 659–678. Suche in Google Scholar

[32] J.-H. Zhang and W.-Y. Yu, Jordan derivations of triangular algebras, Linear Algebra Appl. 419 (2006), no. 1, 251–255. 10.1016/j.laa.2006.04.015Suche in Google Scholar

[33] F. Zhao and C. Li, Nonlinear * -Jordan triple derivations on von Neumann algebras, Math. Slovaca 68 (2018), no. 1, 163–170. 10.1515/ms-2017-0089Suche in Google Scholar
Received: 2023-12-21
Revised: 2024-06-15
Accepted: 2024-07-09
Published Online: 2024-11-14
Published in Print: 2025-06-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Some results in prime rings involving endomorphisms
  3. On the generalized fractional Laplace–Bessel operator
  4. Smoothness of solutions of differential equations of constant strength in Roumieu spaces
  5. Rings additively generated by certain periodic elements
  6. Rings whose clean elements are uniquely strongly clean
  7. New norm inequalities for Jensen’s gap of analytic functions in Banach algebras with applications
  8. On a nonlinear general eigenvalue problem in Musielak–Orlicz spaces
  9. Characterization of multiplicative Jordan n-higher derivations on unital rings
  10. The multiplicative Lie algebra on general linear groups
  11. The optimal control problem for systems of integro-differential equations with finite and infinite horizon
  12. Algebraic dependence of the Gauss maps on minimal surfaces immersed in ℝ n+1
  13. Nonlinear mixed Jordan-type derivations on *-algebras
  14. SG pseudo-differential operators in Dunkl setting
  15. Exploring the properties of multivariable Hermite polynomials in relation to Apostol-type Frobenius–Genocchi polynomials
  16. The binary products of algebras with genetic realization
  17. A relationship between the structure of a quotient ring and * -identities involving pair of derivations
https://doi.org/10.1515/gmj-2024-2062
Schlagwörter für diesen Artikel
Derivation; Jordan derivation; unital ring

Artikel in diesem Heft

  1. Frontmatter
  2. Some results in prime rings involving endomorphisms
  3. On the generalized fractional Laplace–Bessel operator
  4. Smoothness of solutions of differential equations of constant strength in Roumieu spaces
  5. Rings additively generated by certain periodic elements
  6. Rings whose clean elements are uniquely strongly clean
  7. New norm inequalities for Jensen’s gap of analytic functions in Banach algebras with applications
  8. On a nonlinear general eigenvalue problem in Musielak–Orlicz spaces
  9. Characterization of multiplicative Jordan n-higher derivations on unital rings
  10. The multiplicative Lie algebra on general linear groups
  11. The optimal control problem for systems of integro-differential equations with finite and infinite horizon
  12. Algebraic dependence of the Gauss maps on minimal surfaces immersed in ℝ n+1
  13. Nonlinear mixed Jordan-type derivations on *-algebras
  14. SG pseudo-differential operators in Dunkl setting
  15. Exploring the properties of multivariable Hermite polynomials in relation to Apostol-type Frobenius–Genocchi polynomials
  16. The binary products of algebras with genetic realization
  17. A relationship between the structure of a quotient ring and * -identities involving pair of derivations
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