Derivations with power values on multilinear polynomials

Document Type: Regular Paper

Author

School of Mathematics and Finance, Chuzhou University, China

Abstract

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.

Keywords