### Posner’s Second Theorem and some Related Annihilating Conditions on Lie Ideals

#### Abstract

Let $R$ be a non-commutative prime ring of characteristic different

from $2$ with Utumi quotient ring $U$ and extended centroid $C$, $L$

a non-central Lie ideal of $R$, $F$ and $G$ two non-zero generalized

derivations of $R$. If $[F(u),u]G(u)=0$ for all $u\in L$, then one

of the following holds: (a) there exists $\lambda \in C$ such that $F(x)=\lambda x$, for all $x\in R$; (b) $R\subseteq M_2(C)$, the ring of $2\times 2$ matrices over $C$, and

there exist $a \in U$ and $\lambda \in C$ such that

$F(x)=ax+xa+\lambda x$, for all $x\in R$.

from $2$ with Utumi quotient ring $U$ and extended centroid $C$, $L$

a non-central Lie ideal of $R$, $F$ and $G$ two non-zero generalized

derivations of $R$. If $[F(u),u]G(u)=0$ for all $u\in L$, then one

of the following holds: (a) there exists $\lambda \in C$ such that $F(x)=\lambda x$, for all $x\in R$; (b) $R\subseteq M_2(C)$, the ring of $2\times 2$ matrices over $C$, and

there exist $a \in U$ and $\lambda \in C$ such that

$F(x)=ax+xa+\lambda x$, for all $x\in R$.

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