### Abstract

Let R be a prime ring with the extended centroid C, L a noncommutative Lie ideal of R and g a nonzero b-generalized derivation of R. For x,y R, let [x,y] = xy - yx. We prove that if [[⋯[[g(xn0),xn1],xn2],...],xnk] = 0 for all x L, where n0,n1,...,nk are fixed positive integers, then there exists λ C such that g(x) = λx for all x R except when R ⊂ M2(F), the 2 × 2 matrix ring over a field F. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.]

Original language | English |
---|---|

Article number | 1850046 |

Journal | Journal of Algebra and its Applications |

Volume | 17 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2018 Mar 1 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Applied Mathematics

### Cite this

*Journal of Algebra and its Applications*,

*17*(3), [1850046]. https://doi.org/10.1142/S0219498818500469

}

*Journal of Algebra and its Applications*, vol. 17, no. 3, 1850046. https://doi.org/10.1142/S0219498818500469

**An Engel condition with b -generalized derivations for Lie ideals.** / Liau, Pao Kuei; Liu, Cheng Kai.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An Engel condition with b -generalized derivations for Lie ideals

AU - Liau, Pao Kuei

AU - Liu, Cheng Kai

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Let R be a prime ring with the extended centroid C, L a noncommutative Lie ideal of R and g a nonzero b-generalized derivation of R. For x,y R, let [x,y] = xy - yx. We prove that if [[⋯[[g(xn0),xn1],xn2],...],xnk] = 0 for all x L, where n0,n1,...,nk are fixed positive integers, then there exists λ C such that g(x) = λx for all x R except when R ⊂ M2(F), the 2 × 2 matrix ring over a field F. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.]

AB - Let R be a prime ring with the extended centroid C, L a noncommutative Lie ideal of R and g a nonzero b-generalized derivation of R. For x,y R, let [x,y] = xy - yx. We prove that if [[⋯[[g(xn0),xn1],xn2],...],xnk] = 0 for all x L, where n0,n1,...,nk are fixed positive integers, then there exists λ C such that g(x) = λx for all x R except when R ⊂ M2(F), the 2 × 2 matrix ring over a field F. The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 75-80, Skew derivations and Engel conditions, Comm. Algebra 42 (2014), 139-152.]

UR - http://www.scopus.com/inward/record.url?scp=85017010381&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85017010381&partnerID=8YFLogxK

U2 - 10.1142/S0219498818500469

DO - 10.1142/S0219498818500469

M3 - Article

AN - SCOPUS:85017010381

VL - 17

JO - Journal of Algebra and Its Applications

JF - Journal of Algebra and Its Applications

SN - 0219-4988

IS - 3

M1 - 1850046

ER -