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Characterizing Jordan Derivations of Matrix Rings Through Zero Products

  • Hoger Ghahramani EMAIL logo
Published/Copyright: February 9, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 65 Issue 6

Abstract

Let Mn(R) be the ring of all n × n matrices over a unital ring R, let M be a 2-torsion free unital Mn(R)-bimodule and let D: Mn(R)→M be an additive map. We prove that if D(a)b + aD(b) + D(b)a + bD(a) = 0 whenever a,b ∈ Mn(R) are such that ab = ba = 0, then D(a) = δ(a) + aD(1), where δ : Mn(R)→M is a derivation and D(1) lies in the centre of M. It is also shown that D is a generalized derivation if and only if D(a)b + aD(b) + D(b)a + bD(a) − aD(1)b − bD(1)a = 0 whenever ab = ba = 0. We apply this results to provide that any (generalized) Jordan derivation from Mn(R) into a 2-torsion free Mn(R)-bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if φ: Mn(R) → Mn(R) is an additive map satisfying φ(ab+ba) = aφ(b) + φ(b)a (a,b ∈ Mn(R)), then φ(a) = aφ(1) for all a ∈ Mn(R), where φ(1) lies in the centre of Mn(R). By applying this result we obtain that every Jordan derivation of the trivial extension of Mn(R) by Mn(R) is a derivation.

Keywords: Jordan derivation; generalized Jordan derivation; matrix ring

References

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Received: 2012-2-29
Accepted: 2013-3-20
Published Online: 2016-2-9
Published in Print: 2015-12-1

Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0089
Keywords for this article
Jordan derivation; generalized Jordan derivation; matrix ring

Articles in the same Issue

  1. Radius, Diameter and the Degree Sequence of a Graph
  2. On Primary Ideals in Posets
  3. Characterization of the Set of Regular Elements in Ordered Semigroups
  4. Tame Automorphisms with Multidegrees in the Form of Arithmetic Progressions
  5. A Result Concerning Additive Mappings in Semiprime Rings
  6. Characterizing Jordan Derivations of Matrix Rings Through Zero Products
  7. Existence Results for Impulsive Nonlinear Fractional Differential Equations With Nonlocal Boundary Conditions
  8. The Radon-Nikodym Property and the Limit Average Range
  9. A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting
  10. On Booth Lemniscate and Hadamard Product of Analytic Functions
  11. Recursion Formulas for Srivastava Hypergeometric Functions
  12. Regularly Varying Solutions of Half-Linear Diffferential Equations with Retarded and Advanced Arguments
  13. Singular Degenerate Differential Operators and Applications
  14. The Interior Euler-Lagrange Operator in Field Theory
  15. On Selections of Set-Valued Maps Satisfying Some Inclusions in a Single Variable
  16. The Family F of Permutations of ℕ
  17. Summation Process of Positive Linear Operators in Two-Dimensional Weighted Spaces
  18. On Iλ-Statistical Convergence in Locally Solid Riesz Spaces
  19. Some Norm one Functions of the Volterra Operator
  20. Some Results on Absolute Retractivity of the Fixed Points Set of KS-Multifunctions
  21. Convexity in the Khalimsky Plane
  22. Natural Boundary Conditions in Geometric Calculus of Variations
  23. Exponential Inequalities for Bounded Random Variables
  24. Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of φ-Mixing Sequences
  25. Second Order Riemannian Mechanics
  26. Further Remarks on an Order for Quantum Observables
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