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 2 R and n > 1 a fixed integer. In this paper,we show that ifR is prime, then d = 0 or R is commutative. If R is semiprime, thend maps R into its center. Moreover in semiprime case letA = O(R) be the orthogonal completion of R and B = B(C ) be the Boolian ring ofC, where C is the extended centroid of R . Then there exists an idempotente 2 B such that eA is a commutative ring and dinduces a zero derivation on (1− e)A .