Siberian Mathematical Journal

, Volume 53, Issue 6, pp 1051–1060

# Centralizers of generalized derivations on multilinear polynomials in prime rings

• L. Carini
• V. De Filippis
Article

## Abstract

Let R be a prime ring of characteristic different from 2, with Utumi quotient ring U and extended centroid C, δ a nonzero derivation of R, G a nonzero generalized derivation of R, and f(x 1, …, x n ) a noncentral multilinear polynomial over C. If δ(G(f(r 1, …, r n ))f(r 1, …, r n )) = 0 for all r 1, …, r n R, then f(x 1, …, x n )2 is central-valued on R. Moreover there exists aU such that G(x) = ax for all xR and δ is an inner derivation of R such that δ(a) = 0.

## Keywords

prime ring differential identities generalized derivations

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## References

1. 1.
Lee T. K. and Shiue W. K., “Derivations cocentralizing polynomials,” Taiwanese J. Math., 2, No. 4, 457–467 (1998).
2. 2.
Carini L. and De Filippis V., “On some central differential identities in prime rings,” Aligarh Bull. Math., 25, No. 2, 51–58 (2006).
3. 3.
Demir C. and Argac N., “Prime rings with generalized derivations on right ideals,” Algebra Colloq., 18, No. 1, 987–998 (2011).
4. 4.
De Filippis V., “A product of two generalized derivations on polynomials in prime rings,” Collect. Math., 61, No. 3, 303–322 (2010).
5. 5.
Chuang C. L., “The additive subgroup generated by a polynomial,” Israel J. Math., 59, No. 1, 98–106 (1987).
6. 6.
Leron U., “Nil and power central polynomials in rings,” Trans. Amer. Math. Soc., 202, 97–103 (1975).
7. 7.
Beidar K. I., “Rings with generalized identities,” Moscow Univ. Math. Bull., 33, No. 4, 53–58 (1978).
8. 8.
Chuang C. L., “GPI’s having coefficients in Utumi quotient rings,” Proc. Amer. Math. Soc., 103, No. 3, 723–728 (1988).
9. 9.
Erickson T. S., Martindale III W. S., and Osborn J. M., “Prime nonassociative algebras,” Pacific J. Math., 60, No. 1, 49–63 (1975).
10. 10.
Martindale III W. S., “Prime rings satisfying a generalized polynomial identity,” J. Algebra, 12, No. 4, 576–584 (1969).
11. 11.
Lanski C., “An Engel condition with derivation,” Proc. Amer. Math. Soc., 118, No. 3, 731–734 (1993).
12. 12.
Jacobson N., Structure of Rings, Amer. Math. Soc., Providence (1964).Google Scholar
13. 13.
Wong T. L., “Derivations with power central values on multilinear polynomials,” Algebra Colloq., 3, No. 4, 369–378 (1996).
14. 14.
Faith C. and Utumi Y., “On a new proof of Litoff’s theorem,” Acta Math. Acad. Sci. Hungar., 14, 369–371 (1963).
15. 15.
Lee T. K., “Generalized derivations of left faithful rings,” Comm. Algebra, 27, No. 8, 4057–4073 (1999).
16. 16.
Kharchenko V. K., “Differential identities of prime rings,” Algebra and Logic, 17, No. 2, 155–168 (1978).