A note on power values of derivation in prime and semiprime rings

Abstract

Let R be a ring with derivation d, such that (d(xy))^n =(d(x))^n(d(y))^n for all x,y in R and n>1 is a fixed integer. In this paper, we show that if R is a prime, then d = 0 or R is a commutative. If R is a semiprime, then d maps R in to its center. Moreover, in semiprime case let A = O(R) be the orthogonal completion of R and B = B(C) be the Boolian ring of C, where C is the extended centroid of R, then there exists an idempotent e in B such that eA is commutative ring and d induce a zero derivation on (1-e)A.Comment: 8 page