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 ∈ R and n > 1 a fixed integer. In this paper, we show that if R is prime, then d = 0 or R is commutative. If R is semiprime, then d maps R into 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 ∈ B such that eA is a commutative ring and d induces a zero derivation on (1 − e)A