### Abstract

Let R be a prime ring with a nonzero derivation d and let f (X _{1},...,X_{t}) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x_{1},...,x_{t})), f(x1,...,x_{t})]_{n} = 0 for all x_{i} ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X_{1},..., X_{t}) is centrally valued on R unless char R = 2 and dim_{C} RC = 4. We prove a more generalized version by replacing R with a left ideal.

Original language | English |
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Pages (from-to) | 719-725 |

Number of pages | 7 |

Journal | Communications in Algebra |

Volume | 33 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2005 Dec 5 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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*Communications in Algebra*, vol. 33, no. 3, pp. 719-725. https://doi.org/10.1081/AGB-200049880

**Derivations with Engel and annihilator conditions on multilinear polynomials.** / Liu, Cheng Kai.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Derivations with Engel and annihilator conditions on multilinear polynomials

AU - Liu, Cheng Kai

PY - 2005/12/5

Y1 - 2005/12/5

N2 - Let R be a prime ring with a nonzero derivation d and let f (X 1,...,Xt) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x1,...,xt)), f(x1,...,xt)]n = 0 for all xi ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X1,..., Xt) is centrally valued on R unless char R = 2 and dimC RC = 4. We prove a more generalized version by replacing R with a left ideal.

AB - Let R be a prime ring with a nonzero derivation d and let f (X 1,...,Xt) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x1,...,xt)), f(x1,...,xt)]n = 0 for all xi ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X1,..., Xt) is centrally valued on R unless char R = 2 and dimC RC = 4. We prove a more generalized version by replacing R with a left ideal.

UR - http://www.scopus.com/inward/record.url?scp=28044435416&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=28044435416&partnerID=8YFLogxK

U2 - 10.1081/AGB-200049880

DO - 10.1081/AGB-200049880

M3 - Article

AN - SCOPUS:28044435416

VL - 33

SP - 719

EP - 725

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 3

ER -