# Article

Full entry | PDF   (0.2 MB)
Keywords:
generalized skew derivation; Lie ideal; prime ring
Summary:
Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\geq 1$ a fixed positive integer. Let $\alpha$ be an automorphism of the ring $R$. An additive map $D\colon R\to R$ is called an $\alpha$-derivation (or a skew derivation) on $R$ if $D(xy)=D(x)y+\alpha (x)D(y)$ for all $x,y\in R$. An additive mapping $F\colon R\to R$ is called a generalized $\alpha$-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F(xy)=F(x)y+\alpha (x)D(y)$ for all $x,y\in R$. We prove that, if $F$ is a nonzero generalized skew derivation of $R$ such that $F(x)\* [F(x),x]^n = 0$ for any $x\in L$, then either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$, or $R\subseteq M_2(C)$ and there exist $a\in Q_r$ and $\lambda \in C$ such that $F(x)=ax+xa+\lambda x$ for any $x\in R$.
References:
[1] Beidar, K. I., III., W. S. Martindale, Mikhalev, A. V.: Rings with Generalized Identities. Pure and Applied Mathematics 196 Marcel Dekker, New York (1996). MR 1368853 | Zbl 0847.16001
[2] Carini, L., Filippis, V. De: Commutators with power central values on a Lie ideal. Pac. J. Math. 193 (2000), 269-278. DOI 10.2140/pjm.2000.193.269 | MR 1755818 | Zbl 1009.16034
[3] Carini, L., Filippis, V. De, Scudo, G.: Power-commuting generalized skew derivations in prime rings. Mediterr. J. Math. 13 (2016), 53-64. DOI 10.1007/s00009-014-0493-z | MR 3456907 | Zbl 1342.16039
[4] Chang, J.-C.: On the identity {$h(x)=af(x)+g(x)b$}. Taiwanese J. Math. 7 (2003), 103-113. DOI 10.11650/twjm/1500407520 | MR 1961042 | Zbl 1048.16018
[5] Chuang, C. L.: Differential identities with automorphisms and antiautomorphisms. II. J. Algebra 160 (1993), 130-171. DOI 10.1006/jabr.1993.1181 | MR 1237081
[6] Chuang, C.-L.: Differential identities with automorphisms and antiautomorphisms. I. J. Algebra 149 (1992), 371-404. DOI 10.1016/0021-8693(92)90023-F | MR 1172436
[7] Chuang, C.-L.: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103 (1988), 723-728. DOI 10.1090/S0002-9939-1988-0947646-4 | MR 0947646 | Zbl 0656.16006
[8] Chuang, C.-L., Lee, T.-K.: Identities with a single skew derivation. J. Algebra 288 (2005), 59-77. DOI 10.1016/j.jalgebra.2003.12.032 | MR 2138371 | Zbl 1073.16021
[9] Filippis, V. De: Generalized derivations and commutators with nilpotent values on Lie ideals. Tamsui Oxf. J. Math. Sci. 22 (2006), 167-175. MR 2285443 | Zbl 1133.16022
[10] Filippis, V. De, Vincenzo, O. M. Di: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40 (2012), 1918-1932. DOI 10.1080/00927872.2011.553859 | MR 2945689 | Zbl 1258.16043
[11] Filippis, V. De, Scudo, G.: Strong commutativity and Engel condition preserving maps in prime and semiprime rings. Linear Multilinear Algebra 61 (2013), 917-938. DOI 10.1080/03081087.2012.716433 | MR 3175336 | Zbl 1281.16045
[12] Dhara, B., Kar, S., Mondal, S.: Generalized derivations on Lie ideals in prime rings. Czech. Math. J. 65 (140) (2015), 179-190. DOI 10.1007/s10587-015-0167-4 | MR 3336032 | Zbl 1337.16033
[13] Vincenzo, O. M. Di: On the {$n$}-th centralizer of a Lie ideal. Boll. Unione Mat. Ital., A Ser. (7) 3 (1989), 77-85. MR 0990089 | Zbl 0692.16022
[14] Herstein, I. N.: Topics in Ring Theory. Chicago Lectures in Mathematics The University of Chicago Press, Chicago (1969). MR 0271135 | Zbl 0232.16001
[15] Jacobson, N.: Structure of Rings. Colloquium Publications 37. Amer. Math. Soc. Providence (1964). MR 0222106
[16] Lanski, C.: An Engel condition with derivation. Proc. Am. Math. Soc. 118 (1993), 731-734. DOI 10.1090/S0002-9939-1993-1132851-9 | MR 1132851 | Zbl 0821.16037
[17] Lanski, C.: Differential identities, Lie ideals, and Posner's theorems. Pac. J. Math. 134 (1988), 275-297. DOI 10.2140/pjm.1988.134.275 | MR 0961236 | Zbl 0614.16028
[18] Lanski, C., Montgomery, S.: Lie structure of prime rings of characteristic 2. Pac. J. Math. 42 (1972), 117-136. DOI 10.2140/pjm.1972.42.117 | MR 0323839
[19] Lee, T.-K.: Generalized skew derivations characterized by acting on zero products. Pac. J. Math. 216 (2004), 293-301. DOI 10.2140/pjm.2004.216.293 | MR 2094547 | Zbl 1078.16038
[20] Lee, T.-K., Liu, K.-S.: Generalized skew derivations with algebraic values of bounded degree. Houston J. Math. 39 (2013), 733-740. MR 3126322 | Zbl 1285.16038
[21] III., W. S. Martindale: Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969), 576-584. DOI 10.1016/0021-8693(69)90029-5 | MR 0238897 | Zbl 0175.03102
[22] Posner, E. C.: Derivations in prime rings. Proc. Am. Math. Soc. 8 (1957), 1093-1100. DOI 10.1090/S0002-9939-1957-0095863-0 | MR 0095863
[23] Rowen, L. H.: Polynomial Identities in Ring Theory. Pure and Applied Math. 84 Academic Press, New York (1980). MR 0576061 | Zbl 0461.16001
[24] Wang, Y.: Power-centralizing automorphisms of Lie ideals in prime rings. Commun. Algebra 34 (2006), 609-615. DOI 10.1080/00927870500387812 | MR 2211941 | Zbl 1093.16020
[25] Wong, T.-L.: Derivations with power-central values on multilinear polynomials. Algebra Colloq. 3 (1996), 369-378. MR 1422975 | Zbl 0864.16031

Partner of