GENERALIZED SKEW DERIVATIONS AS JORDAN HOMOMORPHISMS ON MULTILINEAR POLYNOMIALS

Title & Authors
GENERALIZED SKEW DERIVATIONS AS JORDAN HOMOMORPHISMS ON MULTILINEAR POLYNOMIALS
De Filippis, Vincenzo;

Abstract
Let $\small{\mathcal{R}}$ be a prime ring of characteristic different from 2, $\small{\mathcal{Q}_r}$ be its right Martindale quotient ring and $\small{\mathcal{C}}$ be its extended centroid. Suppose that $\small{\mathcal{G}}$ is a nonzero generalized skew derivation of $\small{\mathcal{R}}$, $\small{{\alpha}}$ is the associated automorphism of $\small{\mathcal{G}}$, f($\small{x_1}$, $\small{{\cdots}}$, $\small{x_n}$) is a non-central multilinear polynomial over $\small{\mathcal{C}}$ with n non-commuting variables and $\small{\mathcal{S}=\{f(r_1,{\cdots},r_n)\left|r_1,{\cdots},r_n{\in}\mathcal{R}\}}$. If $\small{\mathcal{G}}$ acts as a Jordan homomorphism on $\small{\mathcal{S}}$, then either $\small{\mathcal{G}(x)=x}$ for all $\small{x{\in}\mathcal{R}}$, or $\small{\mathcal{G}={\alpha}}$.
Keywords
polynomial identity;generalized skew derivation;prime ring;
Language
English
Cited by
References
1.
E. Albas and N. Argac, Generalized derivations of prime rings, Algebra Colloq. 11 (2004), no. 3, 399-410.

2.
S. Ali and S. Huang, On generalized Jordan (${\alpha}$, ${\beta}$)-derivations that act as homomor-phisms or anti-homomorphisms, J. Algebra Computat. Appl. 1 (2011), no. 1, 13-19.

3.
A. Ali and D. Kumar, Derivation which acts as a homomorphism or as an anti-homomorphism in a prime ring, Int. Math. Forum 2 (2007), no. 21-24, 1105-1110.

4.
A. Ali and D. Kumar, Generalized derivations as homomorphisms or as anti-homomorphisms in a prime ring, Hacet. J. Math. Stat. 38 (2009), no. 1, 17-20.

5.
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.

6.
K. I. Beidar, W. S. Martindale III, and A. V. Mikhalev, Rings with Generalized Identities, Pure and Applied Math., Dekker, New York, 1996.

7.
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.

8.
J. -C. Chang, On the identity h(x) = af(x) + g(x)b, Taiwanese J. Math. 7 (2003), no. 1, 103-113.

9.
J. -C. Chang, Generalized skew derivations with annihilating Engel conditions, Taiwanese J. Math. 12 (2008), no. 7, 1641-1650.

10.
J. -C. Chang, Generalized skew derivations with nilpotent values on Lie ideals, Monatsh. Math. 161 (2010), no. 2, 155-160.

11.
H.-W. Cheng and F. Wei, Generalized skew derivations of rings, Adv. Math. (China) 35 (2006), no. 2, 237-243.

12.
C. -L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723-728.

13.
C. -L. Chuang, Differential identities with automorphisms and antiautomorphisms I, J. Algebra 149 (1992), no. 2, 371-404.

14.
C. -L. Chuang, Differential identities with automorphisms and antiautomorphisms II, J. Algebra 160 (1993), no. 1, 130-171.

15.
C. -L. Chuang and T.-K. Lee, Rings with annihilator conditions on multilinear polynomials, Chinese J. Math. 24 (1996), no. 2, 177-185.

16.
C. -L. Chuang and T.-K. Lee, Identities with a single skew derivation, J. Algebra 288 (2005), no. 1, 59-77.

17.
V. De Filippis, Generalized derivations as Jordan homomorphisms on Lie ideals and right ideals, Acta Math. Sin. 25 (2009), no. 12, 1965-1974.

18.
V. De Filippis, A products of two generalized derivations on polynomials in prime rings, Collect. Math. 61 (2010), no. 3, 303-322.

19.
V. De Filippis, Annihilators of power values of generalized derivations on multilinear polynomials, Bull. Aust. Math. Soc. 80 (2009), no. 2, 217-232.

20.
I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago 1969.

21.
N. Jacobson, Structure of Rings, Amer. Math. Soc., Providence, RI, 1964.

22.
V. K. Kahrchenko, Generalized identities with automorphisms, Algebra and Logic 14 (1975), 132-148.

23.
V. K. Kahrchenko, Differential identities of prime rings, Algebra Log. 17 (1978), 155-168.

24.
T.-K. Lee, Derivations with invertible values on a multilinear polynomial, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1077-1083.

25.
T.-K. Lee, Generalized skew derivations characterized by acting on zero products, Pacific J. Math. 216 (2004), no. 2, 293-301.

26.
U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97-103.

27.
K.-S Liu, Differential identities and constants of algebraic automorphisms in prime rings, Ph.D. Thesis, National Taiwan University, 2006.

28.
S.-J. Luo, Posner's theorems with skew derivations, Master Thesis, National Taiwan University, 2007.

29.
W. S.Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584.

30.
L. Oukhtite, S. Salhi, and L. Taoufiq, ${\sigma}$-Lie ideals with derivations as homomorphisms and anti-homomorphisms, Int. J. Algebra 1 (2007), no. 5-8, 235-239.

31.
N. Rehman, On generalized derivations as homomorphisms and anti-homomorphisms, Glas. Mat. III ser.39 N.1 (2004), 27-30.

32.
G. Scudo, Generalized derivations acting as Lie on polynomials in prime rings, South-east Asian Bull. Math. 38 (2014), 563-572.

33.
Y. Wang, Generalized derivations with power-central values on multilinear polynomials, Algebra Colloq. 13 (2006), no. 3, 405-410.

34.
Y. Wang and H. You, Derivations as homomorphisms or anti-homomorphisms on Lie ideals, Acta Math. Sin. 23 (2007), no. 6, 1149-1152.

35.
T.-L. Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq. 3 (1996), no. 4, 369-378.

36.
X. Xu, J. Ma, and F. Niu, Compositions, derivations and polynomials, Indian J. Pure Appl. Math. 44 (2013), no. 4, 543-556.