Homoderivations in Prime Rings

The study consists of two parts. first part shows that if $h_{1}(x)h_{2}(y)=h_{3}(x)h_{4}(y)$, for all $x,y\in R$, then $ h_{1}=h_{3}$ and $h_{2}=h_{4}$. Here, $h_{1},h_{2},h_{3},$ $h_{4}$ are zero-power valued non-zero homoderivations a prime ring $R$. Moreover, this provide an explanation related to $h_{1}$ $h_{2}$ satisfying the condition $ah_{1}+h_{2}b=0$. second $L\subseteq Z$ one following conditions is satisfied: $i. h(L)=(0)$, ii. h(L)\subseteq Z$, $iii. h(xy)=xy$, L$, $iv. h(xy)=yx$, or $v. h([x,y])=0$, L$. $R$ with characteristic other than $2$, $h$ homoderivation $R$, $L$ square closed Lie ideal