Derivations with power values on multilinear polynomials

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A polynomial 1 2 ( , , , ) n f X X X is called multilinear if it is homogeneous and linear in every one of its variables. In the present paper our objective is to prove the following result: Let R be a prime K-algebra over a commutative ring K with unity and let 1 2 ( , , , ) n f X X X be a multilinear polynomial over K. Suppose that d is a nonzero derivation on R such that 1 2 1 2 ( , , , ) ( , , , ) s t df x x x n  f x x xn for all 1 2 , , , n x x x R, where s,t are fixed positive integers. Then 1 2 ( , , , ) n f X X X is central-valued on R . We also examine the case R which is a semiprime K-algebra.