Abstract
Let R be a prime ring with extended centroid C, g a nonzero generalized derivation of R, f (x 1,..., x n) a multilinear polynomial over C, I a nonzero right ideal of R.
If [g(f(r 1,..., r n)), f(r 1,..., r n)] = 0, for all r 1, ..., r n ∈ I, then either g(x) = ax, with (a − γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds:
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(i)
f(x 1,..., x n) is central valued in eRCe
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(ii)
g(x) = cx + xb, where (c+b+α)e = 0, for α ∈ C, and f (x 1,..., x n)2 is central valued in eRCe
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(iii)
char(R) = 2 and s 4(x 1, x 2, x 3, x 4) is an identity for eRCe.
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De Filippis, V. An Engel condition with generalized derivations on multilinear polynomials. Isr. J. Math. 162, 93–108 (2007). https://doi.org/10.1007/s11856-007-0090-y
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DOI: https://doi.org/10.1007/s11856-007-0090-y