COMPOSITION OF DERIVATIONS ON MULTILINEAR POLYNOMIALS IN PRIME RINGS

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2,with extended centroid C, d and delta non-zero derivations of R,f(X1, .. ,Xn) a multilinear polynomial over K, I a non-zero right ideal of R. If delta d(f(r1,...,rn))-f(r1, .. ,rn))=O, for all r1,...,rn epsilon I. then one of the following holds:(i) there exists e2= e in the socle of RC such that IC=eRC and f(x1, ... ,xn) is central-valued on eRCe.(ii) delta is the inner derivation induced by an element a and d is the inner derivation induced by theelement b such that a I=bI=O and ba= -a.