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Lie ideals with generalized derivations and derivations of semiprime rings

Sögütcü Emine Koç, Rehman Nadeem ur, Dernek Eda

Fasciculi Mathematici

2023

nr 66

107--120

Let R be a 2-torsion free semiprime ring, L a square-closed Lie ideal of R, ø be a derivation of R and α be an automorphism of R. We will demonstrate in this study that ø(L) = (0), and so ø is a zero map on L if any one of the following holds for all r r ∈ L: (i) ø(r)r = 0 (or r ø(r) = 0) (ii) ø(r) r + r (α (r) - r) = 0, (iii) The mapping r → ø (r) + α (r) is commuting on L. Moreover, if any one of the following are satisfied for two generalized derivations (F, ø) and (H, ξ) of R, then ø is a commuting map on L: (iv) F (r) F (s) - H(rs) ∈ Z (R), (v) F (rs) = ± H (rs), (vi) F (rs) = ±H (sr), for all r, s ∈ L.