NILPOTENCY OF DERIVATIONS IN PRIME-RINGS

In 1957, E. C. Posner proved that if lambda and delta are derivations of a prime ring R, characteristic R not equivalent to 2, then lambda d elta = 0 implies either lambda = 0 or delta = 0. We extend this well-k nown result by showing that, without any characteristic restriction, l ambda delta(m) = 0 implies either lambda = 0 or delta(4m-1) = 0. We al so prove that lambda(n) delta = 0 implies either delta(2) = 0 or lambd a(12n-9) = 0. In the case where lambda(n) delta(m) = 0, we show that i f lambda and delta commute, then at least one of the derivations must be nilpotent.