Some identities involving multiplicative semiderivations on ideals

Abstract

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 ∈ R. In the present paper, we shall prove that [x, d(x)] = 0, for all x ∈ I if any of the followings holds: i) d(xy) ± xy ∈ Z, ii) d(xy) ± yx ∈ Z, iii) d(x)d(y) ± xy ∈ Z, iv) d(xy) ± d(x)d(y) ∈ Z, viii) d(xy) ± d(y)d(x) ∈ Z, for all x, y ∈ I. Also, we show that R must be commutative if d(I) ⊆ Z.