### Abstract

Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f(X_{1}, . . ., X_{t}) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d(f(x_{1},...,x_{t}))f(x_{1},...,x_{t})-f(x_{1},...,x_{t})δ(f(x_{1},...,x_{t}))ε C for all x_{1},...,x_{t} ε λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds:(1) f(X_{1}, . . ., X_{t}) is central-valued on eRCe;(2) λ(d + δ)(λ) = 0 and f (X_{1}, . . ., X_{t})^{2} is central-valued on eRCe;(3) char R = 2 and eRCe satisfies st_{4}(X_{1}, X_{2}, X_{3}, X_{4}), the standard polynomial identity of depsiee 4.

Original language | English |
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Pages (from-to) | 297-311 |

Number of pages | 15 |

Journal | Monatshefte fur Mathematik |

Volume | 162 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2011 Mar 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Monatshefte fur Mathematik*, vol. 162, no. 3, pp. 297-311. https://doi.org/10.1007/s00605-009-0179-y

**Derivations cocentralizing multilinear polynomials on left ideals.** / Liu, Cheng-Kai.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Derivations cocentralizing multilinear polynomials on left ideals

AU - Liu, Cheng-Kai

PY - 2011/3/1

Y1 - 2011/3/1

N2 - Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f(X1, . . ., Xt) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d(f(x1,...,xt))f(x1,...,xt)-f(x1,...,xt)δ(f(x1,...,xt))ε C for all x1,...,xt ε λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds:(1) f(X1, . . ., Xt) is central-valued on eRCe;(2) λ(d + δ)(λ) = 0 and f (X1, . . ., Xt)2 is central-valued on eRCe;(3) char R = 2 and eRCe satisfies st4(X1, X2, X3, X4), the standard polynomial identity of depsiee 4.

AB - Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f(X1, . . ., Xt) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d(f(x1,...,xt))f(x1,...,xt)-f(x1,...,xt)δ(f(x1,...,xt))ε C for all x1,...,xt ε λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds:(1) f(X1, . . ., Xt) is central-valued on eRCe;(2) λ(d + δ)(λ) = 0 and f (X1, . . ., Xt)2 is central-valued on eRCe;(3) char R = 2 and eRCe satisfies st4(X1, X2, X3, X4), the standard polynomial identity of depsiee 4.

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UR - http://www.scopus.com/inward/citedby.url?scp=79952192939&partnerID=8YFLogxK

U2 - 10.1007/s00605-009-0179-y

DO - 10.1007/s00605-009-0179-y

M3 - Article

AN - SCOPUS:79952192939

VL - 162

SP - 297

EP - 311

JO - Monatshefte fur Mathematik

JF - Monatshefte fur Mathematik

SN - 0026-9255

IS - 3

ER -