# A Note on Posner's Theorem with Generalized Derivations on Lie Ideals

### V. De Filippis

Università di Messina, Italy### M.S. Tammam El-Sayid

Beni Suef University, Egypt

## Abstract

Let $R$ be a prime ring of characteristic different from 2, $Z(R)$ its center, $U$ its Utumi quotient ring, $C$ its extended centroid, $G$ a non-zero generalized derivation of $R$, $L$ a non-central Lie ideal of $R$. We prove that if $[[G(u),u],G(u)]∈Z(R)$ for all $u∈L$ then one of the following holds:

- there exists $α∈C$ such that $G(x)=αx$, for all $x∈R$;
- satisfies the standard identity $s_{4}(x_{1},…,x_{4})$ and there exist $a∈U$, $α∈C$ such that $G(x)=ax+xa+αx$, for all $x∈R$.

## Cite this article

V. De Filippis, M.S. Tammam El-Sayid, A Note on Posner's Theorem with Generalized Derivations on Lie Ideals. Rend. Sem. Mat. Univ. Padova 122 (2009), pp. 55–64

DOI 10.4171/RSMUP/122-5