Let R be a prime ring of characteristic different from 2 with right Martindale quotient ring Q and extended centroid C. Let further k ≥ 1 be a fixed integer, f (x1, . . . , xn) a multilinear polynomial over C which is not central-valued on R. If F : R → R is a nonzero generalized skew derivation of R such that [F(f (r1, . . . , rn)), f (r1, . . . , rn)]k = 0 for all r1, . . . , rn ∈ R, then either there exists λ ∈ C such that F(x) = λx for all x ∈ R, or one of the following holds: (a) char(R) = p > 0 and f (x1, . . . , xn)ps is central-valued on R for a suitable s ≥ 0; (b) there exist a, b ∈ Q with a − b ∈ C such that F(x) = ax + xb for all x ∈ R, and f (x1, . . . , xn)2 is central-valued on R. © 2017, © 2017 Informa UK Limited, trading as Taylor & Francis Group.

### An Engel condition with generalized skew derivations on multilinear polynomials

#### Abstract

Let R be a prime ring of characteristic different from 2 with right Martindale quotient ring Q and extended centroid C. Let further k ≥ 1 be a fixed integer, f (x1, . . . , xn) a multilinear polynomial over C which is not central-valued on R. If F : R → R is a nonzero generalized skew derivation of R such that [F(f (r1, . . . , rn)), f (r1, . . . , rn)]k = 0 for all r1, . . . , rn ∈ R, then either there exists λ ∈ C such that F(x) = λx for all x ∈ R, or one of the following holds: (a) char(R) = p > 0 and f (x1, . . . , xn)ps is central-valued on R for a suitable s ≥ 0; (b) there exist a, b ∈ Q with a − b ∈ C such that F(x) = ax + xb for all x ∈ R, and f (x1, . . . , xn)2 is central-valued on R. © 2017, © 2017 Informa UK Limited, trading as Taylor & Francis Group.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11570/3133751`
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