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POWER CENTRAL VALUES OF DERIVATIONS ON MULTILINEAR POLYNOMIALS

Chi-Ming Chang and 張志銘
Taiwanese Journal of Mathematics
Vol. 7, No. 2 (June 2003), pp. 329-338
Stable URL: http://www.jstor.org/stable/43833445
Page Count: 10
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
POWER CENTRAL VALUES OF DERIVATIONS ON MULTILINEAR POLYNOMIALS
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Abstract

Let R be a prime ring with extended centroid C, ρ a nonzero right ideal of R, d a nonzero derivation of R, f(X₁, ... , Xt) a multilinear polynomial over C, a ϵ R and n a fixed positive integer. (I) If ad(f(x₁,... , xt))n = 0 (d(f(x₁,... , xt))n a = 0) for all x₁, ... , xt ϵ ρ, then either aρ = 0 (a = 0 resp.), d(ρ)ρ = 0 or ρC = eRC for some idempotent e in the socle of RC such that f(X₁, ... , Xt) is central-valued on eRCe. (II) If ad(f(x₁,... , xt))n ϵ C(d(f(x₁ ,... ,xt))na ϵ C) for all x₁, ..., xt ϵ ρ and ad(f(y₁,... ,yt))n ≠ 0 (d(f(y₁ ,... , yt))n a ≠ 0) for some y₁,... , yt ϵ ρ, then either f(p) ρ = 0 or f (X₁,... , Xt) is centralvalued on RC unless dimC RC = 4.

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