γ-Lie structures in γ-prime gamma rings with derivations

Let $m$ be a $gamma$-prime weak nobusawa $gamma $-ring and $dneq 0$ be a $k$-derivation of $m$ such that $kleft( gamma right) =0$ and $u$ be a $gamma$-lie ideal of $m$. in this paper, we introduce definitions of $gamma$-subring, $gamma$-ideal, $gamma$-prime $gamma$-ring and $gamma$-lie ideal of m and prove that if $unsubseteq c_{gamma}$, $char$m$neq2$ and $d^3neq0$, then the $gamma$-subring generated by $d(u)$ contains a nonzero ideal of $m$. we also prove that if $[u,d(u)]_{gamma}in c_{gamma}$ for all $uin u$, then $u$ is contained in the $gamma$-center of $m$ when char$mneq2$ or $3$. and if $[u,d(u)]_{gamma}in c_{gamma}$ for all $uin u$ and $u$ is also a $gamma$-subring, then $u$ is $gamma$-commutative when char$m=2$.