Let $R$ $R$ be a prime ring of characteristic different from $2$ $2$ , $Q_{r}$ $Q_{r}$ be the right Martindale quotient ring of $R$ $R$ , $C = Z (Q_{r})$ $C = Z (Q_{r})$ the extended centroid of $R$ $R$ , $L$ $L$ be a noncentral Lie ideal of $R$ $R$ , $H$ $H$ and $G$ $G$ be two nonzero $b$ $b$ -generalized derivations of $R$ $R$ . Suppose there exist fixed integers $m, n \geq 1$ $m, n \geq 1$ such that $H (u^{m}) u^{n} - u^{n} G (u^{m}) = 0$ $H (u^{m}) u^{n} - u^{n} G (u^{m}) = 0$ , for all $u \in L$ $u \in L$ , then either $R$ $R$ satisfies the standard identity $s_{4} (x_{1}, \dots, x_{4})$ $s_{4} (x_{1}, \dots, x_{4})$ or there is $a^{'} \in Q_{r}$ such that $H (x) = x a^{'}$ , $G (x) = a^{'} x$ , for any $x \in R$ , and one of the following holds:

^(a)	there exists $a^{'} \in C$ ,
^(b)	${[x_{1}, x_{2}]}^{m}$ and ${[x_{1}, x_{2}]}^{n}$ are $C$ -linearly dependent.

Then, in the second part of the paper we prove a similar result in the case $H$ and $G$ are generalized skew derivations of $R$ such that $H (x^{m}) x^{n} - x^{n} G (x^{m}) = 0$ , for all $x \in R$ .

Communicated by G. Scudo

Keywords:

AMSC: 16N60, 16W25

References

1. K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with Generalized Identities, Pure and Applied Mathematics (Dekker, New York, 1996). Google Scholar
2. M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993) 385–394. Crossref, Web of Science, Google Scholar
3. L. Carini and V. De Filippis, Identities with generalized derivations on prime rings and Banach algebras, Algebra Colloq. 19 (2012) 971–986. Link, Web of Science, Google Scholar
4. J. C. Chang, On the identity $h (x) = a f (x) + g (x) b$ , Taiwanese J. Math. 7 (2003) 103–113. Crossref, Web of Science, Google Scholar
5. C. L. Chuang, On invariant additive subgroups, Israel J. Math. 57 (1987) 116–128. Crossref, Web of Science, Google Scholar
6. C. L. Chuang, GPI’s having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988) 723–728. Crossref, Google Scholar
7. C. L. Chuang, Differential identities with automorphisms and antiautomorphisms I, J. Algebra 149 (1992) 371–404. Crossref, Web of Science, Google Scholar
8. C. L. Chuang, Differential identities with automorphisms and antiautomorphisms II, J. Algebra 160 (1993) 130–171. Crossref, Web of Science, Google Scholar
9. C. L. Chuang, M.-C. Chou and C.-K. Liu, Skew derivations with annihilating Engel conditions, Publ. Math. Debrecen 68 (2006) 161–170. Crossref, Web of Science, Google Scholar
10. C. L. Chuang and T. K. Lee, Identities with a single skew derivation, J. Algebra 288 (2005) 59–77. Crossref, Web of Science, Google Scholar
11. O. M. Di Vincenzo, On the $n$ -th centralizers of a Lie ideal, Boll. Un. Mat. Ital. 3 (1989) 77–85. Google Scholar
12. T. S. Erickson, W. S. Martindale III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975) 49–63. Crossref, Web of Science, Google Scholar
13. V. De Filippis and O. M. Di Vincenzo, Generalized skew derivations and nilpotent values on Lie ideals, Algebra Colloq. 26 (2019) 589–614. Link, Web of Science, Google Scholar
14. V. De Filippis and G. Scudo, Strong commutativity and Engel condition preserving maps in prime and semiprime rings, Linear Multilinear Algebra 61 (2013) 917–938. Crossref, Web of Science, Google Scholar
15. I. N. Herstein, Topics in Ring Theory (University of Chicago Press, Chicago, 1969). Google Scholar
16. N. Jacobson, Structure of Rings, Amer. Math. Soc. Coll. Pub. 37 (1964) ix+299 pp. Google Scholar
17. V. K. Kharchenko, Differential identities of prime rings, Algebra Logic 17 (1978) 155–168. Crossref, Google Scholar
18. V. K. Kharchenko, Authomorphisms and Derivations of Associative Rings, Mathematics Applications (Soviet Series) (Kluwer Academic Publisher Group, Dordrecht, 1991). Crossref, Google Scholar
19. M. T. Koşan and T. K. Lee, b-Generalized derivations of semiprime rings having nilpotent values, J. Aust. Math. Soc. 96 (2014) 326–337. Crossref, Web of Science, Google Scholar
20. C. Lanski, S. Montgomery, Lie structure of prime rings of characteristic $2$ , Pacific J. Math. 42 (1972) 117–136. Crossref, Web of Science, Google Scholar
21. P. H. Lee and T. L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sinica 23 (1995) 1–5. Google Scholar
22. T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999) 4057–4073. Crossref, Web of Science, Google Scholar
23. T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992) 27–38. Google Scholar
24. T. K. Lee and Y. Zhou, An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317. Link, Web of Science, Google Scholar
25. W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969) 576–584. Crossref, Web of Science, Google Scholar
26. E. C. Pošner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957) 1093–1100. Crossref, Google Scholar
27. T. L. Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq. 3 (1996) 369–378. Google Scholar