Derivations on multilinear polynomials in semiprime rings

Abstract

Let R, he a 2-torsion free semiprime K-algebra with unity, d a non-zero derivation of Rand a non-zero multilinear polynomial over K. Suppose that, for every is zero or invertible in R. Then either R. is a division ring, or is a central polynomial for R, or , for every , the left Utumi quotient ring of R, that is there exists a central idempotent e of Usuch that d vanishes identically on eUand is central in (1-e)U. Moreover the last conclusion holds if and only if , for every ri.