Derivations with invertible or nilpotent values on a multilinear polynomial

Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X-1,...,X-t) a multilinear polynomial not central-valued on R. Suppose d(f(x(1),...,x(t))) is either invertible or nilpotent for all x(1),...,x(t) in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2x2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier. 1991 Mathematics Subject Classification: primary 16W25, secondary 16R50, 16N60, 16U80.