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$M=2$.