Article Sidebar

Submitted
January 5, 2024
Accepted
May 15, 2024
Published
September 11, 2024
Download
PDF
Statistic
Read Counter : 200 Download : 115

Downloads

Download data is not yet available.

Main Article Content

Abstract

In this overview article, we provide a historical account on derivations, Jordan derivations, (α, β)-derivations, left derivations, pre-derivations, homoderivations, nilpotent derivations, and other variants, drawing from the contributions of multiple researchers. Additionally, we delve into recent findings and suggest potential avenues for future investigation in this area. Furthermore, we offer pertinent examples to illustrate that the assumptions underlying various results are indeed necessary and not redundant.

Keywords

(α, β)-derivations bi-derivation Jordan derivation invertible derivation n-Jordan homoderivation

Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

How to Cite
Ali, S., Rafiquee, N. N., & Varshney, V. (2024). CERTAIN TYPES OF DERIVATIONS IN RINGS: A SURVEY. Journal of the Indonesian Mathematical Society, 30(2), 256–306. https://doi.org/10.22342/jims.30.2.1623.256-306
Download Citation
Endnote/Zotero/Mendeley (RIS)
BibTeX
Crossref
Scopus
Google Scholar
Europe PMC

References

  1. Ali, A. and Shujat, F., On symmetric biderivations of semiprime rings, Contemporary Ring Theory, (2012), 196-208.
  2. Ali, S., Study of derivations and commutativity of rings, Ph. D. Thesis, Aligarh Muslim University, (2002).
  3. Ali, S, On generalized left derivations in rings and Banach algebras, Aequat. Math., 81 (2011), 209-226.
  4. Ali, S. and Dar, N. A., On *-centralizing mappings in rings with involution, Georgian Math. J., 21(1) (2014), 25-28.
  5. Ali, S., Alsuraiheed, T. M., Khan, M. S., Abdioglu. C.,Ayedh, M., Rafiquee, N. N., Posner’s theorem and *-centralizing derivations on prime ideals with applications, Mathematics, 11(4) (2003), 3117.
  6. Ancochea, G., Le th´eor`eme de yon staundt en g´eometrie projective quaternionienne, Journal f¨ur die reine und angewandte Mathematik, 184 (1942), 192-198.
  7. Ancochea, G.,On semi-automorphisms of division algebras, Annals of Mathematics, 48 (1947), 147-153
  8. Argac, N., Kaya, A. and Kisir, A. (σ, τ )-derivations in prime rings, Math. J. Okayama Univ (1987).
  9. Ashraf, M., On symmetric bi-derivations in rings, Rend. Istit. Mat. Univ. Trieste, 31 (1999), 25-36.
  10. Ashraf, M., Ali, S., On generalized Jordan left derivations in rings, Bull. Korean Math. Soc., 45(2) (2008), 253-261.
  11. Ashraf, M., Khan, A., Jamal, M. R., Traces of permuting generalized n-derivations of rings, Miskolc Math. Notes, 19(2) (2018), 731-740.
  12. Ashraf, M., Jamal, M. R., Traces of permuting n-additive maps and permuting n-derivations of rings, Mediterr. J. Math., 11 (2014), 287-297.
  13. Ashraf, M., Jamal, M. R., Mozumder, M. R., On the traces of certain classes of permuting mappings in rings, Georgian Math. J., 23(1) (2016), 15-23.
  14. Ashraf, M., Parveen, N., Jamal, M. R., Traces of permuting n-derivations and commutativity of rings, Southeast Asian Bull. Math., 38 (2014), 321-332.
  15. Ashraf, M., Rehman, N., On (σ, τ )-derivations in prime rings, Archivum Mathematicum, 38(4) (2002), 259-264.
  16. Ashraf, M. and Rehman, N., On Lie ideals and Jordan left derivations of prime rings, Arch. Math. (Brno), 36 (2000), 201-206.
  17. Ashraf, M., Rehman, N. and Ali, S., On Jordan left derivations of Lie ideals in prime rings, Southeast Asian Bull. Math., 25 (2001), 379-382.
  18. Ashraf, M., Rehman, N. and Quadri, M. A., On (σ, τ )-derivations in certain classes of rings, Rad. Math., 9 (1999), 187-192.
  19. Ashraf, M., Rehman, N., Ali, S., On Jordan Left Derivations of Lie Ideals in Prime Rings, Southeast Asian Bull. Math., 25 (2001), 379-382.
  20. Ashraf, M., Siddeeque, M. A., On *-n-derivations in rings with involution, Georgian Math. J., 22(1) (2015), 9-18.
  21. Atteya, M., Commutativity with Derivations of Semiprime Rings, Discuss. Math. Gen. Al-gebra Appl., (2020).
  22. Awtar, R., Lie ideals and Jordan derivations of prime rings, Proc. Amer. Math. Soc., 90(1) (1984), 9-14.
  23. Aydin, N., Kaya, K., Some generalizations in prime rings with (σ, τ )-derivation, Turkish J. of Math., 16 (1992).
  24. Bajo, I., Lie algebras admitting non-singular prederivations, Indag. Math. (N.S.), 8 (4) (1997), 433-437.
  25. Banning, R.,Mathieu, M., Commutativity preserving mappings on semiprime rings, Comm. Algebra, 25 (1997) , 247-265.
  26. Beidar, K. I., Chebotar, M. A., On functional identities and d-free subsets of rings I, Comm. Algebra, 28 (2000), 3925-3951.
  27. Belkadi, S., Ali, S. & Taoufiq, L., On nilpotent homoderivations in prime rings, Comm. Algebra, 51(9) (2023), 4044-4053.
  28. Belkadi, S., Ali, S., Taoufiq, L., On n-Jordan homoderivations in rings, Georgian Math. J., (2023), DOI:10.1515/gmj-2023-2065
  29. Bell, H. E., Some commutativity results for rings with two-variable constraints, Proc. Amer. Math. Soc., 53(2) (1975), 280-284.
  30. Bell, H. E., Daif, M. N., Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci., 15(1) (1992), 205-206.
  31. Bell, H. E., and M. N. Daif, On Commutativity and Strong Commutativity-Preserving Maps, Canad. J. Math., 37(4) (1994), 443-447.
  32. Bell, H. E. and M. N. Daif, On derivations and commutativity in prime rings, Acta Math. Hungar., 66 (1995), 337-343.
  33. Bell, H. E., Kappe, L. C., Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar., 53 (1989), 339-346.
  34. Bell, H. E. and W. S. Martindale, Centralizing mappings of semi-prime rings, Canad. Math. Bull., 30 (1987), 92-101.
  35. Benkovi˘c, D., Jordan derivations and antiderivations on triangular matrices, Linear Algebra Appl., 397 (2005), 235-244.
  36. Benkovi˘c, D., Lie derivations on triangular matrices, Linear Multilinear Algebra, 55 (2007), 619-629.
  37. Bergen, J., Lie ideals with regular and nilpotent elements and a result on derivations, Rend. Circ. Mat., Palermo (Ser. 2) 33 (1984), 99-108.
  38. Bergen, J., Carini, L., Derivations with invertible values on a Lie ideal, Canad. Math. Bull., 31(1) (1988), 103-110.
  39. Bergen, J., Herstein, I. N. and Lanski, C., Derivations with invertible values, Canad. J. Math. XXXV(2) (1983), 300-310.
  40. Bergen, J., Herstein, I.N. and Kerr, J.W., Lie ideals and derivations of prime rings, J. Algebra, 71 (1) (1981), 259-267.
  41. Blau, P. S., Lie isomorphisms of non-GPI rings with involution, Comm. Algebra, 27 (1999), 2345-2373.
  42. Boucher, D., Ulmer, F., Coding with skew polynomial rings, Journal of Symbolic Computation 44(12) (2009), 1644-1656.
  43. Boucher, D., Ulmer, F., Linear codes using skew polynomials with automorphisms and derivations, Des. Codes and Cryptogr. 70, 405-431.
  44. Bre˘sar, M., Jordan derivations on semiprime rings, Proc. Amer. Math. Soc., 104 (1988), 1003-1006.
  45. Bre˘sar, M., On a generalization of the notion of centralizing mappings, Proc. Amer. Math. Soc., 114 (1992), 641-649.
  46. Bre˘sar, M., Centralizing mappings and derivations in prime rings, J. Algebra, 156 (1993), 385-394.
  47. Bre˘sar, M., Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc., 335 (1993), 525-546.
  48. Bre˘sar, M., Chebotar, M. A., Martindale 3rd W. S., Functional Identities, Birkh¨auser Verlag, Basel (2007).
  49. Bre˘sar, M. and Hvala, B., On additive maps prime rings, Bull. Aust. Math. Soc., 51 (1995), 377-381.
  50. Bre˘sar, M., Martindale, W. S., and Miers, C. R., Centralizing maps in prime rings with involution, J. Algebra., 161 (1993), 342-357.
  51. Bre˘sar, M., Semel, P., Commuting traces of biadditive maps revisited, Comm. Algebra, 31 (2003), 381-388.
  52. Bre˘sar, M., Villena, A. R., The Noncommutative Singer Wermer Conjecture and Φ-Derivations, J. Lond. Math. Soc., 66(3) (2002), 710-720.
  53. Bre˘sar, M., Vukman, J., Jordan derivations on prime rings, Bull. Aust. Math. Soc., 37 (1988), 321-322.
  54. Bre˘sar, M. and Vukman, J., On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1) (1990), 7-16.
  55. Burde, D., Lie algebra prederivations and strongly nilpotent Lie algebras, Comm. Algebra, 30(7) (2002), 3157-3175.
  56. Chang, J. C., (α, β)-derivation with nilpotent values, Chinese Journal of Mathematics, 22(4) (1994), 349-355.
  57. Chang, J. C., A special identity of (α, β)-derivations and its consequences, Taiwanese J. Math., 1(1) (1997), 21-30.
  58. Chuang, C. L., On compositions of derivations of prime rings, Proc. Amer. Math. Soc., 180 (1990), 647-652.
  59. Chuang, C. L. and Lee, T. K., Finite products of derivations in prime rings, Comm. Algebra, 30(5) (2002), 2183-2190
  60. Chuang, C. L. and Lee, T. K., Identities with a single skew derivation, J. Algebra, 288(1) (2005), 59-77.
  61. Cusack, J. M., Jordan derivations on rings, Proc. Amer. Math. Soc., 53(2) (1975), 321-324.
  62. Chung, L. O. and Luh, J., Nilpotency of derivations I, Canad. Math. Bull. Vol., 26(3) 1983, 341-346.
  63. Chung, L. O. and Luh, J., Nilpotency of derivations II, Proc. Amer. Math. Soc., 91(3) (1984), 357-358.
  64. Chung, L. O. and Luh, J., Nilpotency of derivations on an ideal, Proc. Amer. Math. Soc., 90(2) (1984), 211-214.
  65. Chung, L. O., Nil derivations, J. Algebra, 95(1) (1985), 20-30.
  66. Chung, L. O., Kovacs, A. and Luh, J., Algebraic derivations, preprint.
  67. Daif, M. N., Commutativity results for semiprime rings with derivations, Int. J. Math. Sci., 21(3) (1998), 471-474.
  68. Daif, M. N., Bell, H. E., Remarks on derivations on semiprime rings, Int. J. Math. Math. Sci., 15(1) (1992), 205-206.
  69. Deng, Q., On Jordan left derivations, Math. J. Okayama Univ., 34 (1992), 145-147.
  70. Deng, Q., Ashraf, M., On strong commutativity preserving mappings, Results. Math. 30 (1996), 259-263.
  71. Deng, Q., On a conjecture of Vukman, Int. J. Math. Math. Sci., 20(2) (1997), 263-266.
  72. Dixmier, J., Lister, W. G., Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc. 8 (1957), 155-157.
  73. El-Sofy, M. M., Rings with some kinds of mappings, Ph.D. Thesis, (2000) Cairo University, Branch of Fayoum, Cairo, Egypt.
  74. Favre, G., Une alg`ebre de Lie charact`eristiquement nilpotente de dimension 7. CR., Acad. Sci. Paris, s´er. A 274 (1972), 1338-1339.
  75. Felzenszwalb B., Lanski, C., On the centralizers of ideals and nil derivations, J. Algebra 83 (1983), 520-530.
  76. Fo˘sner, A., Baydar, N., Strasek, R., Remarks on Certain Identities with Derivations on Semiprime Rings, Ukrainian Math. J., 66 (2015), 1609-1614.
  77. Fo˘sner, A., Jing, W., A note on Jordan derivations of triangular rings, Aequationes Math., 94 (2020), 277-285.
  78. Giambruno, A., Misso, P., Milies, C. P., Derivations with Invertible Values in rings with Involution, Pac. J. Math., 123(1) (1986), 47-54.
  79. Guven, E., On (σ, τ ) Derivations in Prime Rings, Int. J. Contemp. Math.Sciences, 3(26) (2008), 1289-1293.
  80. Herstein, I. N., A generalization of a theorem of Jacobson, Amer. J. Math., 73 (1951), 756-762.
  81. Herstein, I. N., A generalization of a theorem of Jacobson III, Amer. J. Math., 75 (1953), 106-111.
  82. Herstein, I. N., The structure of a certain class of rings, Amer. J. Math., 75 (1953), 864-871.
  83. Herstein, I. N., Jordan homomorphisms, Trans. Amer. Math. Soc., 81 (1956), 331-351.
  84. Herstein, I.N., Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104-1110.
  85. Herstein, I.N., Sui Commutatori Degli Anelli Semplici, Seminario Mat. e. Fis. di Milano, 33 (1963), 80-86.
  86. Herstein, I.N., A note on derivations, Canad. Math. Bull., 21 (1978), 369-370.
  87. Hongan, M., On a theorem of J. Vukman, Aequationes Math., 52 (1996), 112-115.
  88. Hongan, M., A note on semiprime rings with derivations, Int. J. Math. Math. Sci., 20 (1997), 413-415.
  89. Hosseini, A., Some conditions under which left derivations are zero, Filomat, 31 (2017), 3965-3974.
  90. Hosseini, A. and Fo˘sner, A., The image of Jordan left derivations on algebras, Bol. Soc. Parana. Mat., 38 (2019), 53-61.
  91. Howland, R. A., Lie isomorphisms of derived rings of simple rings, Trans. Amer. Math. Soc., 145 (1969), 383-396.
  92. Hua, L. K., On the automorphisms of a sfield, Proc. Natl. Acad. Sci. USA, 35 (1949), 386-389.
  93. Hua, L. K., A theorem on matrices over an sfield and its applications, Journal of the Chinese Mathematical Society (N.S.), 1 (1951), 110-163.
  94. Jacobs, D. R., Lie derivations on skew elements of simple rings with involution, Ph.D dissertation, University of Massachusetts, (1973).
  95. Jacobson, N. and Rickart, C., Jordan homomorphisms of rings, Trans. Amer. Math. Soc., 69 (1950), 479-502.
  96. Jacobson, N., A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc., (1955), p. 6.
  97. Jensen, D.W., Nilpotency of derivations in prime rings, Proc. Amer. Math. Soc., 123(9) (1995), 2633-2636.
  98. Johnson, B. E., Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Cambd. Philos. Soc., 120 (1996), 455-470.
  99. Jung, Y. S., On left derivations and derivations of Banach algebras, Bull. Korean Math. Soc., 35 (1998), 659-667.
  100. Kaplansky, I., Semi-automorphisms of rings, Duke Math. J., 14 (1947), 521-527.
  101. Kaya, K., On a (σ, τ )-derivations of prime rings, Doga Tr. J. Math. D. C., 12(2)(1988), 46-51.
  102. Kaya, K., G ˜A¼ven, E. and Soyturk, M., On (σ, τ )-derivations of prime rings, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math., (2006) 13.
  103. Kaygorodov, I. B., Popov, Y. S., Alternative algebras admitting derivations with invertible values and invertible derivations, Izv. Math., 78(5) (2014), 922-936.
  104. Khan, M. S., Ali, S., Ayedh, M., Herstein’s theorem for prime ideals in rings with involution involving pair of derivations, Comm. Algebra, 50(6) (2022), 2592-2603.
  105. Khan, M. S., Khan, M. A., Derivations in prime rings and Banach algebra, Int. J. Algebra, 4(27) (2010), 1317-1328.
  106. Kharchenko, V. K., Differential identities of prime rings, Algebra Logika, 17(2) (1978), 220-238; or Algebra Logic 17 (1978), 155-168.
  107. Killiam, E., Lie derivations on skew elements in prime rings with involution, Canad. Math. Bull., 30 (1987), 344-350.
  108. Kim, G. H., A result concerning derivations in noncommutative Banach algebras, Sci. Math. Jpn., (2001) 53.
  109. Kim, K. H., On symmetric bi-derivations and commutativity of prime rings, Gulf J. Math., 7(2), (2019), 22-30.
  110. Ko ˜A§ S¨og¨utc¨u, E., A Characterization of Semiprime Rings with Homoderivations, Journal of New Theory, 42 (2023), 14-28.
  111. Kovacs, A., Nilpotent derivations, Technion preprint series, Mt-453 November (1979).
  112. Krempa, J. and Matczuk, J., On the composition of derivations, Rend. Circ. Mat. Palermo, 33(1984), 441-455.
  113. Lanski, C., Differential identities, Lie ideals, and Posner’s theorems, Pacific. J. Math., 134 (1988), 275-297.
  114. Lanski, C., Derivations nilpotent on subsets of prime rings, Comm. Algebra, 20(5) (1992), 1427-1446.
  115. Lanski, C., Differential identities of prime rings, Kharchenkoˆas theorem and applications, Contemporary Math., 124 (1992), 111-128.
  116. Lanski, C., Lie ideals and derivations in rings with involution, Pacific. J. Math., 69(2) (1997).
  117. Lee, P. H., Lee, T. K., Note on nilpotent derivations, Proc. Amer. Math. Soc., 98(1), (1996).
  118. Lee, T. K., and Liu, C. K., Spectrally bounded ϕˆaderivations on Banach algebras, Proc. Amer. Math. Soc., 133(5) (2005), 1427-1435.
  119. Leerawat, U., and Khun-in, S., On trace of symmetric bi-derivations on rings, Int. J. Math. Comput. Sci., 16, (2021), 743-752.
  120. Leger, G. F., Tˆogˆo, S., Characteristically nilpotent Lie algebras, Duke Math. J., 26 (1959), 623-628.
  121. Lin, J. S., Derivations with Invertible values in Semiprime rings with involution, Chinese Journal of Mathematics, 18(2) (1990), 175-184.
  122. Luh, J., A note on commuting automorphism of prime rings, Amer. Math. Monthly, 77 (1970), 61-62.
  123. Magnin, L., Adjoint and trivial cohomology tables for indecomposable Lie algebras of dimension ≤ 7 over mathbbC, Lecture Notes 1995.
  124. Maksa, G., A remark on symmetric biadditive functions having nonnegative diagonalization, Glasnik Mat., 15(35) (1980), 279-282.
  125. Maksa, G., On the trace of symmetric bi-derivations, C.R. Math. Rep. Acad. Sci. Canada, 9 (1987), 303-307.
  126. Martindale, W. S. 3rd, Lie isomorphisms of primitive rings, Proc. Amer. Math. Soc., 14 (1963), 909-916.
  127. Martindale, W. S. 3rd, Lie derivations of primitive rings, Michigan Math. J., 11 (1964), 183-187.
  128. Martindale, W. S. 3rd, Lie isomorphisms of simple rings, J. Lond. Math. Soc., 44 (1969), 213-221.
  129. Martindale, W. S. 3rd, Lie isomorphisms of prime rings, Trans. Amer. Math. Soc., 142 (1969), 437-455.
  130. Martindale, W. S. 3rd, Lie isomorphisms of the skew elements of a simple ring with involution, J. Algebra, 36 (1975), 408-415.
  131. Martindale, W. S. 3rd, Lie isomorphisms of the skew elements of a prime ring with involution, Comm. Algebra, 4 (1976), 927-977.
  132. Martindale, W. S. 3rd, Lie and Jordan mappings, Contemporary Mathematics, 13 (1982), 173-177.
  133. Martindale, W. S. 3rd, Miers, C. R., On the iterates of derivations of prime rings, Pacific J. Math., 104 (1983), 178-190.
  134. Mathieu, M., Posner’s second theorem deduced from the first, Proc. Amer. Math. Soc., 144(3) (1992), 601-602.
  135. Mathieu, M., Villena, A. R., The structure of Lie derivations on C*-algebras, J. Funct. Anal., 202, 504-525 (2003).
  136. Mayne, J. H., Centralizing automorphisms of prime rings, Canad. Math. Bull., 19 (1976), 113-115.
  137. Mayne, J. H., Ideals and centralizing mappings in prime rings, Proc. Amer. Math. Soc., 86 (1982), 211-212.
  138. Mayne, J. H., Centralizing mappings of prime rings, Canad. Math. Bull., 27 (1984), 122-126.
  139. Melaibari, A., Muthana, N., Al-Kenani, A., Homoderivations on Rings, General Mathematics Notes, 35(1) (2016), 1-8.
  140. Miers, C. R., Lie triple derivations of yon Neumann algebras, Proc. Amer. Math. Soc., 71 (1978), 57-61.
  141. M¨uller, D., Isometries of bi-invariant pseudo-Riemannian metrics on Lie groups, Geom Dedicata, 29 (1989), 65-96.
  142. Nejjar, B., Kacha, A., Mamouni, A. and Oukhtite, L., Commutativity theorems in rings with involution, Comm. Algebra, 45(2) (2017), 698-708.
  143. Park, K. H., On prime and semiprime rings with symmetric n-derivations, J. Chungcheong Math. Soc., 22 (2009), 451-458.
  144. Posner, E. C., Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093-1100.
  145. Rosen, M. P., Lie isomorphisms of a certain class of prime rings, J. Algebra, 89 (1984), 291-317
  146. Sapanci, M., Ozturk, M. A., Jun, Y. B., Symmetric bi-derivations on prime rings, East Asian Math. J., 15(1) (1999), 105-109.
  147. Singer, I. M. and J. Wermer, Derivations on commutative normed algebras, Math. Ann., 129(1) (1955), 260-264.
  148. Smiley, M. F., Jordan homomorphisms onto prime rings, Trans. Amer. Math. Soc., 84 (1957), 426-429.
  149. Swain, G. A., Lie derivations of the skew elements of prime rings with involution, J. Algebra, 184 (1996), 679-704.
  150. Swain, G. A., Blau, P. S., Lie derivations in prime rings with involution, Canad. Math. Bull., 42 (1999), 401-411.
  151. Thomas, M. P., The Image of a Derivation Is Contained in the Radical, Ann. of Math., 128(3), 1988, 435-460.
  152. Villena, A. R., Lie derivations on Banach algebras, J. Algebra, 226 (2000), 390-409.
  153. Vincenzo, D. F. and Fosner, A., A note on skew derivations in prime rings, Bull. Korean Math. Soc., 49 (2012).
  154. Vukman, J., Symmetric hi-derivations on prime and semi-prime rings, Aequationes Math., 38 (1989), 245-254.
  155. Vukman, J., Two results concerning symmetric bi-derivations on prime rings, Aequationes. Math. 40, (1990), 181-189.
  156. Vukman, J., Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc., 109 (1990), 47-52.
  157. Vukman, J., Jordan left derivations on semiprime rings, Math. J. Okayama Univ., 39, (1997).
  158. Vukman, J., On a-derivations of prime and semiprime rings, Demonstr. Math., 38(2) (2005), 283-290.
  159. Vukman, J., Kosi-ulbl, I., On some equations related to derivations in rings, Int. J. Math. Math. Sci., 17 (2005), 2703-2710.
  160. Vukman, J., Identities with products of (α, β)-derivations on prime rings, Demonstr. Math., (2006).
  161. Vukman, J., On left Jordan derivations of rings and Banach algebras, Aequationes. Math., 75, 260-266 (2008).
  162. Wang, Y., Lie (Jordan) derivations of arbitrary triangular algebras, Aequationes Math., 93(6) (2019), 1221-1229.
  163. Wani, B. A., (ϕ, ψ)-derivations on semiprime rings and Banach algebras, Commun. Math., 29(3) (2021), 371-383.
  164. Wolfgang, A. M., A Characterisation of Nilpotent Lie Algebras by Invertible Leibniz-Derivations, Comm. Algebra, 41(7) (2013), 2427-2440.
  165. Zaidi, S. M. A., Ashraf, M., Ali, S., On Jordan ideals and left (θ, θ)-derivations in prime rings, Int. J. Math. Math. Sci., 37 (2004), 1957-1964.