Lie Ideals and Homoderivations in Semiprime Rings

Let S be a 2-torsion free semiprime ring and U be a noncentral square-closed Lie ideal of S. An additive mapping ℏ on S is defined as a homoderivation if ℏ(ab)=ℏ(a)ℏ(b)+ℏ(a)b+aℏ(a) for all a,b∈S. In the present paper, we shall prove that ℏ is a commuting map on U if any one of the following holds: (i)ℏ(a˜1a˜2)+a˜1a˜2∈Z, (ii)ℏ(a˜1a˜2)−a˜1a˜2∈Z, (iii)ℏa˜1∘a˜2=0, (iv)ℏa˜1∘a˜2=a˜1,a˜2, (v)ℏa˜1,a˜2=0, (vi)ℏa˜1,a˜2= (a˜1∘a˜2), (vii)a˜1ℏ(a˜2)±a˜1a˜2∈Z, (viii)a˜1ℏ(a˜2)±a˜2a˜1=0, (ix)a˜1ℏ(a˜2)±a˜1∘a˜2=0, (x)[ℏ(a˜1),a˜2]±a˜1a˜2=0, (xi)[ℏ(a˜1),a˜2]±a˜2a˜1=0, for all a˜1,a˜2∈U, where ℏ is a homoderivation on S.