### Abstract

We apply elementary matrix computations and the theory of differential identities to prove the following: let R be a prime ring with extended centroid C and L a noncommutative Lie ideal of R. Suppose that f:L → R is a map and g is a generalized derivation of R such that [f(x), g(y)]=[x, y] for all x, y ∈ L. Then there exist a nonzero α ∈ C and a map μ: L → C such that g(x)=α x for all x ∈ R and f (x)=α^{-1}x+ μ(x) for all x ∈ L, except when R ⊆ M_{2}(F), the 2 × 2 matrix ring over a field F.

Original language | English |
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Pages (from-to) | 905-915 |

Number of pages | 11 |

Journal | Linear and Multilinear Algebra |

Volume | 59 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2011 Aug 1 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

Liu, C. K., & Liau, P. K. (2011). Strong commutativity preserving generalized derivations on Lie ideals.

*Linear and Multilinear Algebra*,*59*(8), 905-915. https://doi.org/10.1080/03081087.2010.535819