Derivations with Engel and annihilator conditions on multilinear polynomials

Cheng Kai Liu

doi:10.1081/AGB-200049880

Derivations with Engel and annihilator conditions on multilinear polynomials

Cheng Kai Liu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

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(x₁,...,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(X₁,..., 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
Pages (from-to)	719-725
Number of pages	7
Journal	Communications in Algebra
Volume	33
Issue number	3
DOIs	https://doi.org/10.1081/AGB-200049880
Publication status	Published - 2005 Dec 5

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

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10.1081/AGB-200049880

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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.

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