### 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 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Linear and Multilinear Algebra*,

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

}

*Linear and Multilinear Algebra*, vol. 59, no. 8, pp. 905-915. https://doi.org/10.1080/03081087.2010.535819

**Strong commutativity preserving generalized derivations on Lie ideals.** / Liu, Cheng Kai; Liau, Pao Kuei.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Strong commutativity preserving generalized derivations on Lie ideals

AU - Liu, Cheng Kai

AU - Liau, Pao Kuei

PY - 2011/8/1

Y1 - 2011/8/1

N2 - 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-1x+ μ(x) for all x ∈ L, except when R ⊆ M2(F), the 2 × 2 matrix ring over a field F.

AB - 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-1x+ μ(x) for all x ∈ L, except when R ⊆ M2(F), the 2 × 2 matrix ring over a field F.

UR - http://www.scopus.com/inward/record.url?scp=79961150104&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79961150104&partnerID=8YFLogxK

U2 - 10.1080/03081087.2010.535819

DO - 10.1080/03081087.2010.535819

M3 - Article

AN - SCOPUS:79961150104

VL - 59

SP - 905

EP - 915

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 8

ER -