Homoderivations on Rings

Let R be a ring with center Z(R) and let U be a nonzero ideal. An additive mapping h : R ^ R is called a homoderivation on R if h(xy) = h(x)h(y) + h(x)y + xh(y) for all x,y E R. In this paper, we prove the commutativity of R if any of the following conditions is satisfied: (i) [x,y] = [h(x),h(y)] for all x,y E R, (ii) h([x,y]) = 0 for all x,y E U, and (iii) h([x,y]) E Z(R) for all x,y E R.