Homoderivations in Prime Rings

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