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

Abstract

Let $M$ be a $\gamma$-prime weak Nobusawa $\Gamma $-ring and $d\neq 0$ be a $k$-derivation of $M$ such that $k\left( \gamma \right) =0$ 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 $U\nsubseteq C_{\gamma}$, $char$M$\neq2$ and $d^3\neq0$, 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 $u\in U$, then $U$ is contained in the $\gamma$-center of $M$ when char$M\neq2$ or $3$. And if $[u,d(u)]_{\gamma}\in C_{\gamma}$ for all $u\in U$ and $U$ is also a $\gamma$-subring, then $U$ is $\gamma$-commutative when char$M=2$.