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On the traces of certain classes of permuting mappings in rings

  • Mohammad Ashraf EMAIL logo , Malik Rashid Jamal and Muzibur Rahman Mozumder
Published/Copyright: January 19, 2016
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 23 Issue 1

Abstract

Let R be a semiprime ring with center Z and extended centroid C. For a fixed integer n ≥ 2, the trace δ:R→R of a permuting n-additive mapping D:Rn→R is defined as δ(x)=D(x,...,x) for all x ∈ R. The notion of permuting n-derivation was introduced by Park [J. Chungcheong Math. Soc. 22 (2009), no.3, 451–458] as follows: a permuting n-additive mapping Δ:Rn→R is said to be permuting n-derivation if

Δ(x1,x2,⋯,xixi',⋯,xn)=Δ(x1,x2,⋯,xi,⋯,xn)xi'+xiΔ(x1,x2,⋯,xi',⋯,xn)forallxi,xi'∈R.

A permuting n-additive mapping Ω:Rn→R is known to be a permuting generalized n-derivation if there exists a permuting n-derivation Δ:Rn→R such that

Ω(x1,x2,⋯,xixi',⋯,xn)=Ω(x1,x2,⋯,xi,⋯,xn)xi'+xiΔ(x1,x2,⋯,xi',⋯,xn)forallxi,xi'∈R.

The main result of this paper states that if I is a nonzero ideal of a semiprime ring R and Δ:Rn→R is a permuting n-derivation such that Δ(I,...,I)≠{0} and [δ(x),x]=0 for all x ∈ I, where δ is the trace of Δ, then R contains a nonzero central ideal. Furthermore, some related results are also proven.

Keywords: Derivation; generalized derivation; permuting n-derivation; centralizing mapping; prime ring
MSC: 16W25; 16U80

Funding source: Department of Science and Technology of India

Award Identifier / Grant number: INT/SLOVENIA/P-18/2009

The authors would like to express their indebtedness to the referee for valuable remarks and suggestions.

Received: 2013-7-15
Revised: 2015-6-30
Accepted: 2015-7-3
Published Online: 2016-1-19
Published in Print: 2016-3-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. The annihilating-ideal graph of a lattice
  3. On derivations and commutativity of prime rings with involution
  4. On the traces of certain classes of permuting mappings in rings
  5. Measurability properties of certain paradoxical subsets of the real line
  6. Some classical Tauberian theorems for (C,1,1,1) summable triple sequences
  7. Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces
  8. Irrationality measures for almost periodic continued fractions
  9. Elementary volume and measurability properties of additive functions
  10. On (n+1)-Hom-Lie algebras induced by n-Hom-Lie algebras
  11. Generalization of Hermite–Hadamard type inequalities for n-times differentiable functions through preinvexity
  12. Nonlinear iterative algorithms for quasi contraction mapping in modular space
  13. n-dimensional quintic and sextic functional equations and their stabilities in Felbin type spaces
  14. Some remarks on quasi-Frobenius rings
https://doi.org/10.1515/gmj-2015-0051
Keywords for this article
Derivation; generalized derivation; permuting n-derivation; centralizing mapping; prime ring

Articles in the same Issue

  1. Frontmatter
  2. The annihilating-ideal graph of a lattice
  3. On derivations and commutativity of prime rings with involution
  4. On the traces of certain classes of permuting mappings in rings
  5. Measurability properties of certain paradoxical subsets of the real line
  6. Some classical Tauberian theorems for (C,1,1,1) summable triple sequences
  7. Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces
  8. Irrationality measures for almost periodic continued fractions
  9. Elementary volume and measurability properties of additive functions
  10. On (n+1)-Hom-Lie algebras induced by n-Hom-Lie algebras
  11. Generalization of Hermite–Hadamard type inequalities for n-times differentiable functions through preinvexity
  12. Nonlinear iterative algorithms for quasi contraction mapping in modular space
  13. n-dimensional quintic and sextic functional equations and their stabilities in Felbin type spaces
  14. Some remarks on quasi-Frobenius rings
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