Derivations with Power-Central Values on Multilinear Polynomials

摘要

Let R be a prime ring with center Z, let d be a nonzero derivation of R, and let f(X_1, ···, X_t) be a multilinear polynomial not vanishing on R. If d(f(x_1,···, x_t))~n = 0 for all x_1, ···, x_t ∈ R where n = n(x_1, ···, x_t) ≥ 1 is an integer, then d(Z) = 0 and f(X_1, ···, X_t) is central-valued on R provided n is fixed or R contains no nonzero nil one-sided ideals. If d(f(x_1, ···, x_t)) ∈ Z for all x_1, ··· ,x_t ∈ R where n ≥ 1 is a fixed integer, then f(X_1, ···, X_t) is central-valued on R unless R is an order in a 4-dimensional simple algebra.