Co-centralizing generalized derivations acting on multilinear polynomials in prime rings

Document Type : Research Paper


1 Department of Mathematics‎, ‎Belda College‎, ‎Belda‎, ‎Paschim Medinipur‎, ‎721424‎, ‎W.B.‎, ‎India.

2 Department of Mathematics‎, ‎Jadavpur University‎, ‎Kolkata-700032‎, ‎W.B.‎, ‎India.

3 {Department of Mathematics‎, ‎Belda College‎, ‎Belda‎, ‎Paschim Medinipur‎, ‎721424‎, ‎W.B.‎, ‎India.


‎Let $R$ be a noncommutative prime ring of‎ ‎characteristic different from $2$‎, ‎$U$ the Utumi quotient ring of $R$‎, ‎$C$ $(=Z(U))$ the extended centroid‎ ‎of $R$‎. ‎Let $0\neq a\in R$ and $f(x_1,\ldots,x_n)$ a multilinear‎ ‎polynomial over $C$ which is noncentral valued on $R$‎. ‎Suppose‎ ‎that $G$ and $H$ are two nonzero generalized derivations of $R$‎ ‎such that $a(H(f(x))f(x)-f(x)G(f(x)))\in C$ for all‎ ‎$x=(x_1,\ldots,x_n)\in R^n$‎. ‎ one of the following holds‎:
‎$f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exist $b,p,q\in U$ such‎ ‎that $H(x)=px+xb$ for all $x\in R$‎, ‎$G(x)=bx+xq$ for all $x\in R$ with $a(p-q)\in C$;‎
‎there exist $p,q\in U$ such that $H(x)=px+xq$ for all $x\in R$‎,
‎$G(x)=qx$ for all $x\in R$ with $ap=0$;‎
 $f(x_1,\ldots,x_n)^2$ is central valued on $R$ and there exist $q\in U$‎, ‎$\lambda\in C$ and an outer derivation $g$ of $U$‎
‎such that $H(x)=xq+\lambda x-g(x)$ for all $x\in R$‎, ‎$G(x)=qx+g(x)$ for all $x\in R$‎, ‎with $a\in C$;‎
$R$ satisfies $s_4$ and there exist $b,p\in U$ such‎ ‎that $H(x)=px+xb$ for all $x\in R$‎, ‎$G(x)=bx+xp$ for all $x\in R$‎.


Main Subjects

 N. Arga_c and V. De Filippis, Actions of generalized derivations on multilinear polynomials in prime rings, Algebra Colloq. 18 (2011), Special Issue no. 1, 955--964.
K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with Generalized Identities, Monographs and Textbooks in Pure and Applied Mathematics, 196, Marcel Dekker, Inc. New York, 1996.
J. Bergen, I. N. Herstein and J. W. Kerr, Lie ideals and derivations of prime rings, J. Algebra 71 (1981), no. 1, 259--267.
M. Bre_sar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (1993), no. 2, 385--394.
L. Carini, V. De Filippis and B. Dhara, Annihilators on co-commutators with generalized derivations on Lie ideals, Publ. Math. Debrecen 76 (2010), no. 3-4, 395--409.
C. M. Chang and T. K. Lee, Annihilators of power values of derivations in prime rings, Comm. Algebra 26 (1998), no. 7, 2091--2113.
C. L. Chuang and T. K. Lee, Rings with annihilator conditions on multilinear polynomials, Chin. J. Math. 24 (1996), no. 2, 177--185.
C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723--728.
C. L. Chuang, The additive subgroup generated by a polynomial, Israel J. Math. 59 (1987), no. 1, 98--106.
V. De Filippis and B. Dhara, Cocentralizing generalized derivations on multilinear polynomial on right ideals of prime rings, Demonstr. Math. 47 (2014), no. 1, 22--36.
V. De Filippis, B. Dhara and G. Scudo, Annihilating conditions on generalized derivations acting on multilinear polynomials, Georgian Math. J. 20 (2013), no. 4, 641--656.
B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), no. 4, 1147--1166.
V. K. Kharchenko, Differential identity of prime rings, Algebra Logic 17 (1978), no. 2, 220--238, 242--243.
C. Lanski, An engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), no. 3, 731--734.
C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math. 42 (1972), no. 1, 117--136.
P. H. Lee and T. L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sinica, 23 (1995), no. 1, 1--5.
T. K. Lee and W. K. Shiue, Identities with generalized derivations, Comm. Algebra 29 (2001), no. 10, 4437--4450.
T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), no. 8, 4057--4073.
T. K. Lee and W. K. Shiue, Derivations cocentralizing polynomials, Taiwanese J. Math. 2 (1998), no. 4, 457--467.
T. K. Lee, Derivations with invertible values on a multilinear polynomial, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1077--1083.
T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 27--38.
J. S. Lin and C. K. Liu, Generalized derivations with invertible or nilpotent values on multiliear polynomials, Comm. Algebra 34 (2006), no. 2, 633--640.
U. Leron, Nil and power central polynomials in rings, Trans. Amer. Math. Soc. 202 (1975) 97--103.
F. Niu and W. Wu, Annihilators on co-commutators with derivations on Lie ideals in prime rings, Northeast Math. J. 22 (2006), no. 4, 415--424.
E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957) 1093--1100.
L. H. Rowen, Polynomial Identities in Ring Theory, Pure and Applied Mathematics 84, Academic Press, New York, 1980.