Rendiconti del Seminario Matematico della Università di Padova


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Volume 122, 2009, pp. 55–64
DOI: 10.4171/RSMUP/122-5

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

V. De Filippis[1] and M.S. Tammam El-Sayid[2]

(1) D.I.S.I.A., Fac. of Engineering, Università di Messina, Contrada di Dio, 98166, Messina, Italy
(2) Department of Mathematics, Beni Suef University, Beni Suef, Egypt

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 uL then one of the following holds:
1. there exists α ∈ C such that G(x) = αx, for all xR;
2. satisfies the standard identity s4(x1,…,x4) and there exist aU, α ∈ C such that G(x) = ax + xa + αx, for all xR.

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De Filippis V., Tammam El-Sayid M.S.: A Note on Posner's Theorem with Generalized Derivations on Lie Ideals. Rend. Sem. Mat. Univ. Padova 122 (2009), 55-64. doi: 10.4171/RSMUP/122-5