RATHER LARGE SUBSETS OF PRIME AND SEMIPRIME RINGS WITH DERIVATIONS

Abstract

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, with extended centroid C, d and δ non-zero derivations of R, f(x1,..,xn) a polynomial overK. Ifδ(d(f(r1,..,rn))− f(r1,..,rn)) = 0, for all r1,..,rn ∈ R, thenf(x1,..,xn) is central-valued on R. We also examine the case whenR is a two-torsion free semiprime ring, n = 2 andf(x1,x2)=[ x1,x2]k, the k-th commutator in two variables, for k a fixed positive integer.