Acknowledgement
The authors are thankful to the anonymous referee for his/her useful suggestions.
References
- S. Ali and N. A. Dar, On *-centralizing mappings in rings with involution, Georgian Math. J. 21 (2014), no. 1, 25-28. https://doi.org/10.1515/gmj-2014-0006
- S. Ali and N. A. Dar, On centralizers of prime rings with involution, Bull. Iranian Math. Soc. 41 (2015), no. 6, 1465-1475.
- S. Ali, N. A. Dar, and M. A,sci, On derivations and commutativity of prime rings with involution, Georgian Math. J. 23 (2016), no. 1, 9-14. https://doi.org/10.1515/gmj-2015-0016
- S. Ali, M. S. Khan, and M. Ayedh, On central identities equipped with skew Lie product involving generalized derivations, J. King Saud Univ. Sci. 34 (2022), no. 3, Paper No. 101860, 7pp. https://doi.org/10.1016/j.jksus.2022.101860
- 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.
- K. I. Beidar and W. S. Martindale III, On functional identities in prime rings with involution, J. Algebra 203 (1998), no. 2, 491-532. https://doi.org/10.1006/jabr.1997.7285
- K. I. Beidar, W. S. Martindale III, and A. V. Mikhalev, Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, 196, Marcel Dekker, Inc., New York, 1996.
- A. Boua and M. Ashraf, Identities related to generalized derivations in prime *-rings, Georgian Math. J. 28 (2021), no. 2, 193-205. https://doi.org/10.1515/gmj-2019-2056
- 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
- M. Bresar, M. A. Chebotar, and W. S. Martindale III, Functional Identities, Frontiers in Mathematics, Birkhauser Verlag, Basel, 2007.
- M. Bresar and J. Vukman, On some additive mappings in rings with involution, Aequationes Math. 38 (1989), no. 2-3, 178-185. https://doi.org/10.1007/BF01840003
- N. A. Dar and S. Ali, On *-commuting mappings and derivations in rings with involution, Turkish J. Math. 40 (2016), no. 4, 884-894. https://doi.org/10.3906/mat-1508-61
- N. A. Dar and A. N. Khan, Generalized derivations in rings with involution, Algebra Colloq. 24 (2017), no. 3, 393-399. https://doi.org/10.1142/S1005386717000244
- N. J. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III 49 (1955), 19-22.
- S. F. El-Deken and H. Nabiel, Centrally-extended generalized *-derivations on rings with involution, Beitr. Algebra Geom. 60 (2019), no. 2, 217-224. https://doi.org/10.1007/s13366-018-0415-5
- M. P. Eroglu, T.-K. Lee, and J.-H. Lin, Anti-endomorphisms and endomorphisms satisfying an Engel condition, Comm. Algebra 47 (2019), no. 10, 3950-3957. https://doi.org/10.1080/00927872.2019.1572175
- M. Fosner and J. Vukman, Identities with generalized derivations in prime rings, Mediterr. J. Math. 9 (2012), no. 4, 847-863. https://doi.org/10.1007/s00009-011-0158-0
- I. N. Herstein, Rings with Involution, Chicago Lectures in Mathematics, Univ. Chicago Press, Chicago, IL, 1976.
- I. N. Herstein, A note on derivations, Canad. Math. Bull. 21 (1978), no. 3, 369-370. https://doi.org/10.4153/CMB-1978-065-x
- B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), no. 4, 1147-1166. https://doi.org/10.1080/00927879808826190
- M. A. Idrissi and L. Oukhtite, Some commutativity theorems for rings with involution involving generalized derivations, Asian-Eur. J. Math. 12 (2019), no. 1, Paper No. 1950001, 11 pp. https://doi.org/10.1142/S1793557119500013
- C. Lanski, Differential identities, Lie ideals, and Posner's theorems, Pacific J. Math. 134 (1988), no. 2, 275-297. http://projecteuclid.org/euclid.pjm/1102689262 102689262
- T.-K. Lee, Anti-automorphisms satisfying an Engel condition, Comm. Algebra 45 (2017), no. 9, 4030-4036. https://doi.org/10.1080/00927872.2016.1255894
- T.-K. Lee, Commuting anti-homomorphisms, Comm. Algebra 46 (2018), no. 3, 1060-1065. https://doi.org/10.1080/00927872.2017.1335746
- T.-K. Lee, Certain basic functional identities of semiprime rings, Comm. Algebra 47 (2019), no. 1, 17-29. https://doi.org/10.1080/00927872.2018.1439049
- J.-H. Lin, Jordan τ-derivations of prime GPI-rings, Taiwanese J. Math. 24 (2020), no. 5, 1091-1105. https://doi.org/10.11650/tjm/191105
- C.-K. Liu, Additive n-commuting maps on semiprime rings, Proc. Edinb. Math. Soc. (2) 63 (2020), no. 1, 193-216. https://doi.org/10.1017/s001309151900018x
- A. Mamouni, B. Nejjar, and L. Oukhtite, Differential identities on prime rings with involution, J. Algebra Appl. 17 (2018), no. 9, Paper No. 1850163, 11 pp. https://doi.org/10.1142/S0219498818501633
- A. Mamouni, L. Oukhtite, and M. Zerra, Certain algebraic identities on prime rings with involution, Comm. Algebra 49 (2021), no. 7, 2976-2986. https://doi.org/10.1080/00927872.2021.1887203
- 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
- B. Nejjar, A. Kacha, A. Mamouni, and L. Oukhtite, Certain commutativity criteria for rings with involution involving generalized derivations, Georgian Math. J. 27 (2020), no. 1, 133-139. https://doi.org/10.1515/gmj-2018-0010
- L. Oukhtite, Posner's second theorem for Jordan ideals in rings with involution, Expo. Math. 29 (2011), no. 4, 415-419. https://doi.org/10.1016/j.exmath.2011.07.002
- L. Oukhtite and A. Mamouni, Generalized derivations centralizing on Jordan ideals of rings with involution, Turkish J. Math. 38 (2014), no. 2, 225-232. https://doi.org/10.3906/mat-1203-14
- L. Oukhtite and O. A. Zemzami, A study of differential prime rings with involution, Georgian Math. J. 28 (2021), no. 1, 133-139. https://doi.org/10.1515/gmj-2019-2061
- E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. https://doi.org/10.2307/2032686
- L. Rowen, Some results on the center of a ring with polynomial identity, Bull. Amer. Math. Soc. 79 (1973), 219-223. https://doi.org/10.1090/S0002-9904-1973-13162-3
- M. A. Siddeeque, N. Khan, and A. A. Abdullah, Weak Jordan *-derivations of prime rings, J. Algebra Appl. 22 (2023), no. 5, Paper No. 2350105, 34 pp. https://doi.org/10.1142/S0219498823501050
- S. K. Tiwari, R. K. Sharma, and B. Dhara, Identities related to generalized derivation on ideal in prime rings, Beitr. Algebra Geom. 57 (2016), no. 4, 809-821. https://doi.org/10.1007/s13366-015-0262-6
- J. Vukman, Commuting and centralizing mappings in prime rings, Proc. Amer. Math. Soc. 109 (1990), no. 1, 47-52. https://doi.org/10.2307/2048360
- Y. Wang, Power-centralizing automorphisms of Lie ideals in prime rings, Comm. Algebra 34 (2006), no. 2, 609-615. https://doi.org/10.1080/00927870500387812
- O. A. Zemzami, L. Oukhtite, S. Ali, and N. Muthana, On certain classes of generalized derivations, Math. Slovaca 69 (2019), no. 5, 1023-1032. https://doi.org/10.1515/ms-2017-0286