A note on derivations in semiprime rings

We prove in this note the following result. Let n>1 be an integer and let R be an n!-torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D:R→R such that D(xn)=∑j=1nxn−jD(x)xj−1 is fulfilled for all x∈R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associative ring with center Z(R). Given an integer n>1, a ring R is said to be n-torsion-free if for x∈R, nx=0 implies that x=0. Recall that a ring R is prime if for a,b∈R, aRb=(0) implies that either a=0 or b=0, and is semiprime in case aRa=(0) implies that a=0. An additive mapping D:R→R is called a derivation if D(xy)=D(x)y+xD(y) holds for all pairs x,y∈R and is called a Jordan derivation in case D(x2)=D(x)x+xD(x) is fulfilled for all x∈R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein (1957) asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein's result can be found in 1988 by Brešar and Vukman. Cusack (1975) generalized Herstein's result to 2-torsion-free semiprime rings (see also Brešar (1988) for an alternative proof). For some other results concerning derivations on prime and semiprime rings, we refer to Brešar (1989), Vukman (2005), Vukman and Kosi-Ulbl (2005)