[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2015.52.1.191

GENERALIZED SKEW DERIVATIONS AS JORDAN HOMOMORPHISMS ON MULTILINEAR POLYNOMIALS

De Filippis, Vincenzo (Department of Mathematics and Computer Science University of Messina)

Publication Information

Journal of the Korean Mathematical Society / v.52, no.1, 2015 , pp. 191-207 More about this Journal

Abstract

Let $\mathcal{R}$ be a prime ring of characteristic different from 2, $\mathcal{Q}_r$ be its right Martindale quotient ring and $\mathcal{C}$ be its extended centroid. Suppose that $\mathcal{G}$ is a nonzero generalized skew derivation of $\mathcal{R}$ , ${\alpha}$ is the associated automorphism of $\mathcal{G}$ , f( $x_1$ , ${\cdots}$ , $x_n$ ) is a non-central multilinear polynomial over $\mathcal{C}$ with n non-commuting variables and $\mathcal{S}=\{f(r_1,{\cdots},r_n)\left|r_1,{\cdots},r_n{\in}\mathcal{R}\}$ . If $\mathcal{G}$ acts as a Jordan homomorphism on $\mathcal{S}$ , then either $\mathcal{G}(x)=x$ for all $x{\in}\mathcal{R}$ , or $\mathcal{G}={\alpha}$ .

Keywords

polynomial identity; generalized skew derivation; prime ring;

Citations & Related Records

Reference

1	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.
2	E. Albas and N. Argac, Generalized derivations of prime rings, Algebra Colloq. 11 (2004), no. 3, 399-410.
3	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.
4	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. DOI
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. DOI
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. DOI
8	J. -C. Chang, On the identity h(x) = af(x) + g(x)b, Taiwanese J. Math. 7 (2003), no. 1, 103-113. DOI
9	J. -C. Chang, Generalized skew derivations with annihilating Engel conditions, Taiwanese J. Math. 12 (2008), no. 7, 1641-1650. DOI
10	J. -C. Chang, Generalized skew derivations with nilpotent values on Lie ideals, Monatsh. Math. 161 (2010), no. 2, 155-160. DOI
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 and T.-K. Lee, Rings with annihilator conditions on multilinear polynomials, Chinese J. Math. 24 (1996), no. 2, 177-185.
13	C. -L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723-728. DOI ScienceOn
14	C. -L. Chuang, Differential identities with automorphisms and antiautomorphisms I, J. Algebra 149 (1992), no. 2, 371-404. DOI
15	C. -L. Chuang, Differential identities with automorphisms and antiautomorphisms II, J. Algebra 160 (1993), no. 1, 130-171. DOI ScienceOn
16	C. -L. Chuang and T.-K. Lee, Identities with a single skew derivation, J. Algebra 288 (2005), no. 1, 59-77. DOI ScienceOn
17	V. De Filippis, Generalized derivations as Jordan homomorphisms on Lie ideals and right ideals, Acta Math. Sin. 25 (2009), no. 12, 1965-1974. DOI ScienceOn
18	I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago 1969.
19	V. De Filippis, A products of two generalized derivations on polynomials in prime rings, Collect. Math. 61 (2010), no. 3, 303-322. DOI
20	V. De Filippis, Annihilators of power values of generalized derivations on multilinear polynomials, Bull. Aust. Math. Soc. 80 (2009), no. 2, 217-232. DOI
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. DOI
23	V. K. Kahrchenko, Differential identities of prime rings, Algebra Log. 17 (1978), 155-168. DOI
24	T.-K. Lee, Derivations with invertible values on a multilinear polynomial, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1077-1083. DOI ScienceOn
25	S.-J. Luo, Posner's theorems with skew derivations, Master Thesis, National Taiwan University, 2007.
26	T.-K. Lee, Generalized skew derivations characterized by acting on zero products, Pacific J. Math. 216 (2004), no. 2, 293-301. DOI
27	U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975), 97-103. DOI
28	W. S.Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584. DOI
29	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. DOI
30	N. Rehman, On generalized derivations as homomorphisms and anti-homomorphisms, Glas. Mat. III ser.39 N.1 (2004), 27-30. DOI
31	G. Scudo, Generalized derivations acting as Lie on polynomials in prime rings, South-east Asian Bull. Math. 38 (2014), 563-572.
32	Y. Wang, Generalized derivations with power-central values on multilinear polynomials, Algebra Colloq. 13 (2006), no. 3, 405-410. DOI
33	Y. Wang and H. You, Derivations as homomorphisms or anti-homomorphisms on Lie ideals, Acta Math. Sin. 23 (2007), no. 6, 1149-1152. DOI
34	T.-L. Wong, Derivations with power central values on multilinear polynomials, Algebra Colloq. 3 (1996), no. 4, 369-378.
35	X. Xu, J. Ma, and F. Niu, Compositions, derivations and polynomials, Indian J. Pure Appl. Math. 44 (2013), no. 4, 543-556. DOI
36	K.-S Liu, Differential identities and constants of algebraic automorphisms in prime rings, Ph.D. Thesis, National Taiwan University, 2006.