# 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-centralLie ideal of *R*.We prove that if [[*G*(*u*),*u*],*G*(*u*)] ∈ *Z*(*R*) for all *u* ∈ *L* then one of the following holds:

1. there exists α ∈ *C* such that *G*(*x*) = α_x_, for all *x* ∈ *R*;

2. satisfies the standard identity _s_4(_x_1,…,*x_4) and there exist a ∈ U, α ∈ C such that G(x) = ax + xa + α_x*, 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