Rendiconti del Seminario Matematico della Università di Padova
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Published online: 2009-12-31
A Note on Posner's Theorem with Generalized Derivations on Lie IdealsV. De Filippis and M.S. Tammam El-Sayid (1) Università di Messina, Italy
(2) Beni Suef University, 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-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) = αx, for all x ∈ R;
2. satisfies the standard identity s4(x1,…,x4) and there exist a ∈ U, α ∈ C such that G(x) = ax + xa + αx, for all x ∈ R.
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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