# A NOTE ON GENERALIZED DERIVATIONS AS A JORDAN HOMOMORPHISMS

• Received : 2019.04.24
• Accepted : 2020.01.03
• Published : 2020.05.31

#### Abstract

Let R be a prime ring of characteristic different from 2. Suppose that F, G, H and T are generalized derivations of R. Let U be the Utumi quotient ring of R and C be the center of U, called the extended centroid of R and let f(x1, …, xn) be a non central multilinear polynomial over C. If F(f(r1, …, rn))G(f(r1, …, rn)) - f(r1, …, rn)T(f(r1, …, rn)) = H(f(r1, …, rn)2) for all r1, …, rn ∈ R, then we describe all possible forms of F, G, H and T.

#### References

1. N. Argac and V. De Filippis, Actions of generalized derivations on multilinear polynomials in prime rings, Algebra Colloq. 18 (2011), Special Issue no. 1, 955-964. https://doi.org/10.1142/S1005386711000836 https://doi.org/10.1142/S1005386711000836
2. A. Asma, N. Rehman, and A. Shakir, On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar. 101 (2003), no. 1-2, 79-82. https://doi.org/10.1023/B:AMHU.0000003893.61349.98 https://doi.org/10.1023/B:AMHU.0000003893.61349.98
3. 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.
4. H. E. Bell and L.-C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53 (1989), no. 3-4, 339-346. https://doi.org/10.1007/BF01953371 https://doi.org/10.1007/BF01953371
5. M. Bresar, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991), no. 1, 89-93. https://doi.org/10.1017/S0017089500008077 https://doi.org/10.1017/S0017089500008077
6. 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 https://doi.org/10.1006/jabr.1993.1080
7. L. Carini, V. De Filippis, and G. Scudo, Identities with product of generalized derivations of prime rings, Algebra Colloq. 20 (2013), no. 4, 711-720. https://doi.org/10.1142/S1005386713000680 https://doi.org/10.1142/S1005386713000680
8. C.-L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723-728. https://doi.org/10.2307/2046841 https://doi.org/10.1090/S0002-9939-1988-0947646-4
9. B. Dhara, Generalized derivations acting on multilinear polynomials in prime rings, Czechoslovak Math. J. 68(143) (2018), no. 1, 95-119. https://doi.org/10.21136/CMJ.2017.0352-16
10. B. Dhara, S. Sahebi, and V. Rahmani, Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals and right ideals, Math. Slovaca 65 (2015), no. 5, 963-974. https://doi.org/10.1515/ms-2015-0065
11. V. De Filippis and G. Scudo, Generalized derivations which extend the concept of Jordan homomorphism, Publ. Math. Debrecen 86 (2015), no. 1-2, 187-212. https://doi.org/10.5486/PMD.2015.7070 https://doi.org/10.5486/PMD.2015.7070
12. V. De Filippis and O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Comm. Algebra 40 (2012), no. 6, 1918-1932. https://doi.org/10.1080/00927872.2011.553859 https://doi.org/10.1080/00927872.2011.553859
13. N. J. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III 49 (1955), 19-22.
14. T. S. Erickson, W. S. Martindale, 3rd, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49-63. http://projecteuclid.org/euclid.pjm/1102868622 https://doi.org/10.2140/pjm.1975.60.49
15. C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hungar. 14 (1963), 369-371. https://doi.org/10.1007/BF01895723 https://doi.org/10.1007/BF01895723
16. I. N. Herstein, Jordan homomorphisms, Trans. Amer. Math. Soc. 81 (1956), 331-341. https://doi.org/10.2307/1992920 https://doi.org/10.1090/S0002-9947-1956-0076751-6
17. B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), no. 4, 1147-1166. https://doi.org/10.1080/00927879808826190 https://doi.org/10.1080/00927879808826190
18. N. Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, RI, 1964.
19. V. K. Kharchenko, Differential identities of prime rings, Algebra i Logika 17 (1978), no. 2, 220-238, 242-243.
20. T.-K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 27-38.
21. T.-K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), no. 8, 4057-4073. https://doi.org/10.1080/00927879908826682 https://doi.org/10.1080/00927879908826682
22. T.-K. Lee and W.-K. Shiue, Derivations cocentralizing polynomials, Taiwanese J. Math. 2 (1998), no. 4, 457-467. https://doi.org/10.11650/twjm/1500407017 https://doi.org/10.11650/twjm/1500407017
23. T.-K. Lee and W.-K. Shiue, Identities with generalized derivations, Comm. Algebra 29 (2001), no. 10, 4437-4450. https://doi.org/10.1081/AGB-100106767 https://doi.org/10.1081/AGB-100106767
24. U. Leron, Nil and power-central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97-103. https://doi.org/10.2307/1997300 https://doi.org/10.1090/S0002-9947-1975-0354764-6
25. 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 https://doi.org/10.1016/0021-8693(69)90029-5
26. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. https://doi.org/10.2307/2032686 https://doi.org/10.1090/S0002-9939-1957-0095863-0
27. M. F. Smiley, Jordan homomorphisms onto prime rings, Trans. Amer. Math. Soc. 84 (1957), 426-429. https://doi.org/10.2307/1992823 https://doi.org/10.1090/S0002-9947-1957-0083484-X
28. S. K. Tiwari, Generalized derivations with multilinear polynomials in prime rings, Comm. Algebra 46 (2018), no. 12, 5356-5372. https://doi.org/10.1080/00927872.2018.1468899 https://doi.org/10.1080/00927872.2018.1468899
29. 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 https://doi.org/10.1007/s13366-015-0262-6
30. S. K. Tiwari, R. K. Sharma, and B. Dhara, Multiplicative (generalized)-derivation in semiprime rings, Beitr. Algebra Geom. 58 (2017), no. 1, 211-225. https://doi.org/10.1007/s13366-015-0279-x https://doi.org/10.1007/s13366-015-0279-x