Derivations in prime rings

Abstract

Let R R be a ring and d ≠ 0 d \ne 0 a derivation of R R such that d ( x n ) = 0 d({x^n}) = 0 , n = n ( x ) ⩾ 1 n = n(x) \geqslant 1 , for all x ∈ R x \in R . It is shown that if R R is primitive then R R is an infinite field of characteristic p > 0 p > 0 and p | n ( x ) p|n(x) if d ( x ) ≠ 0 d(x) \ne 0 . Moreover, if R R is prime and the set of integers n ( x ) n(x) is bounded, the same conclusion holds.