Composition of Derivations on (semi) Prime I-rings

Abstract

In this paper first we show that the iterationdn ; n 2; of a non zero derivation d on a semi-prime `-ring R is not a derivation, and henced can not be nilpotent. Then we see that if R is prime and d1; ;dn ; n 2; are positive derivations on R with d1 dn a derivation, then at least one of these derivations must be zero. Finally, taking R to be a semiprime f-ring, we show that if for every x 2 R; dn (x) is central, then, under some mild conditions, there is a nonzero inner derivation on R such that (x)d(x) = d(x)(x) = 0.These results generalizes and improves the ones in [3], [8], and [9]. And show that imposing an order on a ring, making it an ordered ring, gives stronger results.