Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

Abstract

Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X₁,...,X _t ) a multilinear polynomial not central-valued on R. Suppose d(f(x₁,...,x _t )) is either invertible or nilpotent for all x₁,...,x _t in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier.