Some identities involving multiplicative semiderivations on ideals

Let R be a prime ring and I be a nonzero ideal of R. A mapping d : R -> R is called a multiplicative semiderivation if there exists a function g : R -> R such that (i) d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and (ii) d(g(x)) = g(d(x)) hold for all x, y is an element of R. In the present paper, we shall prove that [x, d(x)] = 0, for all x is an element of I if any of the followings holds: i) d(xy) +/- xy is an element of Z, H) d(xy) +/- yx is an element of Z, Hi) d(x)d(y) +/- xy is an element of Z, iv) d(xy) +/- d(x)d(y) is an element of Z, viii) d(xy) +/- d(y)d(x) is an element of Z, for all x, y is an element of I. Also, we show that R must be commutative if d(I) subset of Z.