Homoderivations in Prime Rings

The study consists of two parts. The first part shows that if h1(x)h2(y) = h3(x)h4(y), for all x, y ? R, then h1 = h3 and h2 = h4. Here, h1, h2, h3, and h4 are zero- power valued non-zero homoderivations of a prime ring R. Moreover, this study provide an explanation related to h1 and h2 satisfying the condition ah1 + h2b = 0. The second part shows that L ? Z if one of the following conditions is satisfied: i. h(L) = (0), ii. h(L) ? Z, iii. h(xy) = xy, for all x, y ? L, iv. h(xy) = yx, for all x, y ? L, or v. h([x, y]) = 0, and for all x, y ? L. Here, R is a prime ring with a characteristic other than 2, h is a homoderivation of R, and L is a non-zero square closed Lie ideal of R.