A Note on Generalized Derivations with Zero and Invertible Values

Let $R$ be a non-commutative ring of characteristic different from $2$, with center $Z(R)$, Utumi quotient ring $U$ and extended centroid $C$. Let $G$ be a generalized derivation of $R$, $L$ be a non-central Lie ideal of $R$. Here we prove two results concerning the behaviour of $G$ and the structure of $R$. More precisely, if $G(u)u=0$, for all $u\in L$, then $G=0$; if $G$ is an inner generalized derivation and $G(u)u$ is zero or invertible, for all $u\in L$, then either $R$ is a division ring or $R=M_2(D)$ a ring of $2\times 2$ matrices over the disision ring $D$.