Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON 𝜂-GENERALIZED DERIVATIONS IN RINGS WITH JORDAN INVOLUTION

  • Phool Miyan (Department of Mathematics College of Natural and Computational Sciences Haramaya University)
  • Received : 2023.10.18
  • Accepted : 2024.02.01
  • Published : 2024.07.31
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Abstract

Let 𝒦 be a ring. An additive map 𝖚⋄ → 𝖚 is called Jordan involution on 𝒦 if (𝖚⋄)⋄ = 𝖚 and (𝖚𝖛+𝖛𝖚)⋄ = 𝖚⋄𝖛⋄+𝖛⋄𝖚⋄ for all 𝖚, 𝖛 ∈ 𝒦. If Θ is a (non-zero) 𝜂-generalized derivation on 𝒦 associated with a derivation Ω on 𝒦, then it is shown that Θ(𝖚) = 𝛄𝖚 for all u ∈ 𝒦 such that 𝛄 ∈ Ξ and 𝛄2 = 1, whenever Θ possesses [Θ(𝖚), Θ(𝖚⋄)] = [𝖚, 𝖚⋄] for all 𝖚 ∈ 𝒦.

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References

  1. S. Ali, N. A. Dar, and A. N. Khan, On strong commutativity preserving like maps in rings with involution, Miskolc Math. Notes 16 (2015), no. 1, 17-24. https://doi.org/10.18514/mmn.2015.1297
  2. M. Ashraf, A. Ali, and S. Ali, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math. 31 (2007), no. 3, 415-421.
  3. H. E. Bell and M. N. Daif, On commutativity and strong commutativity-preserving maps, Canad. Math. Bull. 37 (1994), no. 4, 443-447. https://doi.org/10.4153/CMB-1994-064-x
  4. H. E. Bell and G. Mason, On derivations in near-rings and rings, Math. J. Okayama Univ. 34 (1992), 135-144.
  5. M. Bresar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), no. 2, 385-394. https://doi.org/10.1006/jabr.1993.1080
  6. M. Bresar, W. S. Martindale III, and C. R. Miers, Centralizing maps in prime rings with involution, J. Algebra 161 (1993), no. 2, 342-357. https://doi.org/10.1006/jabr.1993.1223
  7. M. Bresar and C. R. Miers, Strong commutativity preserving maps of semiprime rings, Canad. Math. Bull. 37 (1994), no. 4, 457-460. https://doi.org/10.4153/CMB-1994-066-4
  8. N. A. Dar and A. N. Khan, Generalized derivations in rings with involution, Algebra Colloq. 24 (2017), no. 3, 393-399.
  9. Q. Deng and M. Ashraf, On strong commutativity preserving mappings, Results Math. 30 (1996), no. 3-4, 259-263. https://doi.org/10.1007/BF03322194 https://doi.org/10.1142/S1005386717000244
  10. A. N. Khan and S. Ali, Involution on prime rings with endomorphisms, AIMS Math. 5 (2020), no. 4, 3274-3283. https://doi.org/10.3934/math.2020210
  11. V. K. Kharchenko, Differential identities of prime rings, Algebra and Logic 17 (1978), 155-168.
  12. M. T. Ko,san and T.-K. Lee, b-generalized derivations of semiprime rings having nilpotent values, J. Aust. Math. Soc. 96 (2014), no. 3, 326-337. https://doi.org/10.1017/S1446788713000670
  13. T.-K. Lee and T.-L. Wong, Nonadditive strong commutativity preserving maps, Comm. Algebra 40 (2012), no. 6, 2213-2218. https://doi.org/10.1080/00927872.2011.578287
  14. J.-S. Lin and C.-K. Liu, Strong commutativity preserving maps on Lie ideals, Linear Algebra Appl. 428 (2008), no. 7, 1601-1609. https://doi.org/10.1016/j.laa.2007.10.006
  15. J.-S. Lin and C.-K. Liu, Strong commutativity preserving maps in prime rings with involution, Linear Algebra Appl. 432 (2010), no. 1, 14-23. https://doi.org/10.1016/j.laa.2009.06.036
  16. C.-K. Liu, Strong commutativity preserving generalized derivations on right ideals, Monatsh. Math. 166 (2012), no. 3-4, 453-465. https://doi.org/10.1007/s00605-010-0281-1
  17. C.-K. Liu, On skew derivations in semiprime rings, Algebr. Represent. Theory 16 (2013), no. 6, 1561-1576. https://doi.org/10.1007/s10468-012-9370-2
  18. C.-K. Liu and P.-K. Liau, Strong commutativity preserving generalized derivations on Lie ideals, Linear Multilinear Algebra 59 (2011), no. 8, 905-915. https://doi.org/10.1080/03081087.2010.535819
  19. W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584. https://doi.org/10.1016/0021-8693(69)90029-5
  20. B. Nejjar, A. Kacha, A. Mamouni, and L. Oukhtite, Commutativity theorems in rings with involution, Comm. Algebra 45 (2017), no. 2, 698-708. https://doi.org/10.1080/00927872.2016.1172629
  21. X. Qi and J. Hou, Strong commutativity preserving maps on triangular rings, Oper. Matrices 6 (2012), no. 1, 147-158. https://doi.org/10.7153/oam-06-10
  22. B. Yood, On Banach algebras with a Jordan involution, Note Mat. 11 (1991), 331-333.