Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 95–119

# Generalized derivations acting on multilinear polynomials in prime rings

Article

## Abstract

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x1,..., x n ) a multilinear polynomial over C which is not central valued on R. If
$$F(f(r))G(f(r)) = H(f(r)^2 )$$
for all r = (r1,..., r n ) ∈ I n , then one of the following conditions holds:
1. (1)

there exist aC and bU such that F(x) = ax, G(x) = xb and H(x) = xab for all xR

2. (2)

there exist a, bU such that F(x) = xa, G(x) = bx and H(x) = abx for all xR, with abC

3. (3)

there exist bC and aU such that F(x) = ax, G(x) = bx and H(x) = abx for all xR

4. (4)
f(x1,..., x n )2 is central valued on R and one of the following conditions holds
1. (a)

there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all xR, with ab = p + p

2. (b)

there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all xR, with p + p’ = ab ∈ C.

## MSC 2010

prime ring derivation generalized derivation extended centroid Utumi quotient ring

16W25 16N60

## References

1. [1]
E. Albaş: Generalized derivations on ideals of prime rings. Miskolc Math. Notes 14 (2013) 3–9.
2. [2]
S. Ali, S. Huang: On generalized Jordan (α, β)-derivations that act as homomorphisms or anti-homomorphisms. J. Algebra Comput. Appl. (electronic only) 1 (2011) 13–19.
3. [3]
N. Argaç, V. De Filippis: Actions of generalized derivations on multilinear polynomials in prime rings. Algebra Colloq. 18, Spec. Iss. 1 (2011), 955–964.
4. [4]
A. Asma, N. Rehman, A. Shakir: On Lie ideals with derivations as homomorphisms and anti-homomorphisms. Acta Math. Hungar 101 (2003) 79–82.
5. [5]
H.E. Bell, L.C. Kappe: Rings in which derivations satisfy certain algebraic conditions. Acta Math. Hung. 53 (1989) 339–346.
6. [6]
J. Bergen, I.N. Herstein, J.W. Keer: Lie ideals and derivations of prime rings. J. Algebra 71 (1981) 259–267.
7. [7]
L. Carini, V. De Filippis, G. Scudo: Identities with product of generalized derivations of prime rings. Algebra Colloq. 20 (2013) 711–720.
8. [8]
C.-L. Chuang: The additive subgroup generated by a polynomial. Isr. J. Math. 59 (1987) 98–106.
9. [9]
C.-L. Chuang: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103 (1988) 723–728.
10. [10]
V. De Filippis: Generalized derivations as Jordan homomorphisms on Lie ideals and right ideals. Acta Math. Sin., Engl. Ser. 25 (2009) 1965–1974.
11. [11]
V. De Filippis, O.M. Di Vincenzo: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40 (2012) 1918–1932.
12. [12]
V. De Filippis, G. Scudo: Generalized derivations which extend the concept of Jordan homomorphism. Publ. Math. 86 (2015) 187–212.
13. [13]
B. Dhara: Derivations with Engel conditions on multilinear polynomials in prime rings. Demonstr. Math. 42 (2009) 467–478.
14. [14]
B. Dhara: Generalized derivations acting as a homomorphism or anti-homomorphism in semiprime rings. Beitr. Algebra Geom. 53 (2012) 203–209.
15. [15]
B. Dhara, S. Huang, A. Pattanayak: Generalized derivations and multilinear polynomials in prime rings. Bull. Malays. Math. Sci. Soc. 36 (2013) 1071–1081.
16. [16]
B. Dhara, N.U. Rehman, M.A. Raza: Lie ideals and action of generalized derivations in rings. Miskolc Math. Notes 16 (2015) 769–779.
17. [17]
B. Dhara, S. Sahebi, V. Rehmani: Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals and right ideals. Math. Slovaca 65 (2015) 963–974.
18. [18]
T. S. Erickson, W. S. Martindale III, J.M. Osborn: Prime nonassociative algebras. Pac. J. Math. 60 (1975) 49–63.
19. [19]
I. Gusić: A note on generalized derivations of prime rings. Glas. Mat., III. Ser. 40 (2005) 47–49.
20. [20]
N. Jacobson: Structure of Rings. American Mathematical Society Colloquium Publications 37, Revised edition American Mathematical Society, Providence, 1956.Google Scholar
21. [21]
V.K. Kharchenko: Differential identities of prime rings. Algebra Logic 17 (1978) 155–168. (In English. Russian original.); translation from Algebra Logika 17 (1978), 220–238.
22. [22]
C. Lanski: Differential identities, Lie ideals, and Posner’s theorems. Pac. J. Math. 134 (1988) 275–297.
23. [23]
C. Lanski: An Engel condition with derivation. Proc. Am. Math. Soc. 118 (1993) 731–734.
24. [24]
T.-K. Lee: Semiprime rings with differential identities. Bull. Inst. Math., Acad. Sin. 20 (1992) 27–38.
25. [25]
T.-K. Lee: Generalized derivations of left faithful rings. Commun. Algebra 27 (1999) 4057–4073.
26. [26]
P.-H. Lee, T.-K. Lee: Derivations with Engel conditions on multilinear polynomials. Proc. Am. Math. Soc. 124 (1996) 2625–2629.
27. [27]
U. Leron: Nil and power central polynomials in rings. Trans. Am. Math. Soc. 202 (1975) 97–103.
28. [28]
W. S. Martindale III: Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969) 576–584.
29. [29]
E.C. Posner: Derivations in prime rings. Proc. Am. Math. Soc. 8 (1957) 1093–1100.
30. [30]
N.U. Rehman: On generalized derivations as homomorphisms and anti-homomorphisms. Glas. Mat., III. Ser. 39 (2004) 27–30.
31. [31]
Y. Wang, H. You: Derivations as homomorphisms or anti-homomorphisms on Lie ideals. Acta Math. Sin., Engl. Ser. 23 (2007) 1149–1152.