Taiwanese Journal of Mathematics


Basudeb Dhara, Vincenzo De Filippis, and Krishna Gopal Pradhan

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Let $R$ be a noncommutative primering with its Utumi ring of quotients $U$, $C=Z(U)$ the extendedcentroid of $R$, $F$ a generalized derivation of $R$ and $I$ anonzero ideal of $R$. Suppose that there exists $0\neq a\in R$ such that $a(F([x,y])^n-[x,y])=0$ for all $x,y \in I$, where $n\geq 1$ is a fixedinteger. Then either $n=1$ and $F(x)=bx$ for all $x\in R$ with$a(b-1)=0$ or $n\geq 2$ and one of the following holds:

1. char $(R)\neq 2$, $R\subseteq M_2(C)$, $F(x)=bx$ for all$x\in R$ with $a(b-1)=0$ (In this case $n$ is an odd integer);

2. char $(R)= 2$, $R\subseteq M_2(C)$ and $F(x)=bx+[c,x]$ forall $x\in R$ with $a(b^n-1)=0$.

Article information

Taiwanese J. Math., Volume 19, Number 3 (2015), 943-952.

First available in Project Euclid: 4 July 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16W25: Derivations, actions of Lie algebras 16N60: Prime and semiprime rings [See also 16D60, 16U10]

prime ring derivation generalized derivation extended centroid Utumi quotient ring


Dhara, Basudeb; De Filippis, Vincenzo; Pradhan, Krishna Gopal. GENERALIZED DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS. Taiwanese J. Math. 19 (2015), no. 3, 943--952. doi:10.11650/tjm.19.2015.4043. https://projecteuclid.org/euclid.twjm/1499133671

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  • N. Argaç and Ç. Dem\.ir, Generalized derivations of prime rings on multilinear polynomials with annihilator conditions, Turk. J. Math., 37 (2013), 231-243.
  • K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math., 196, Marcel Dekker, New York, 1996.
  • C. L. Chuang, GPI's having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103(3) (1988), 723-728.
  • M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Internat. J. Math. Math. Sci., 15(1) (1992), 205-206.
  • V. De Filippis and S. Huang, Generalized derivations on semiprime rings, Bull. Korean Math. Soc., 48(6) (2011), 1253-1259.
  • V. De Filippis, Annihilators of power values of generalized derivations on multilinear polynomials, Bull. Austr. Math. Soc., 80 (2009), 217-232.
  • B. Dhara, V. De Filippis and G. Scudo, Power values of generalized derivations with annihilator conditions in prime rings, Mediterr. J. Math., 10 (2013), 123-135.
  • T. S. Erickson, W. S. Martindale III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60 (1975), 49-63.
  • I. N. Herstein, Center-like elements in prime rings, J. Algebra, 60 (1979), 567-574.
  • I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago, 1969.
  • N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Pub., 37, Amer. Math. Soc., Providence, RI, 1964.
  • V. K. Kharchenko, Differential identity of prime rings, Algebra and Logic, 17 (1978), 155-168.
  • T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra, 27(8) (1999), 4057-4073.
  • T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20(1) (1992), 27-38.
  • W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584.
  • M. A. Quadri, M. S. Khan and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math., 34(9) (2003), 1393-1396.