Geodesic
On symmetric generalized ($\theta,\eta$)-biderivations of prime rings
Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 73-91

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In this paper, we characterize the actions of symmetric generalized (θ,η)-biderivations and generalized left (θ,η)-biderivations on Lie ideals and ideals of a prime ring 𝒜. It is shown that ℒ (nonzero square-closed Lie ideal of 𝒜) ⊆𝒵(𝒜), whenever traces of these derivations satisfy any of the following conditions:(i) ([l_1,l_2])^Δ=0,(ii) (l_1l_2)^Δ∈𝒵(𝒜),(iii) ([l_1,l_2])^Δ=(l_1)^θ∘(l_2)^Δ,(iv) (l_1)^Δ(l_2)^Δ+(l_1)^η(l_2)^θ∈𝒵(𝒜),(v) a_1((l_1)^Δ(l_2)^Δ+(l_1l_2)^θ)=0,(vi) (l_1)^Δ(l_2)^θ+(l_1)^θ(l_2)^Δ=0,(vii) ([l_1,l_2])^Δ+[(l_1)^Δ,l_2]∈𝒵(𝒜),(viii) (l_1l_2)^Δ±(l_1)^θ(l_2)^Δ+ (l_1l_2)^θ∈𝒵(𝒜), ∀ l_1,l_2∈ℒ,where 0≠ a_1∈𝒜 is a fixed element,Δ is a trace of these biadditive mappings andθ, η are automorphisms of 𝒜.
Keywords: Lie ideals, prime rings, generalized $(\theta,\eta)$-biderivations, generalized left $(\theta,\eta)$-biderivations 
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     title = {On symmetric generalized ($\theta,\eta$)-biderivations of prime rings},
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Dadhwal, Madhu; Devi, Geeta. On symmetric generalized ($\theta,\eta$)-biderivations of prime rings. Discussiones Mathematicae. General Algebra and Applications, Tome 44 (2024) no. 1, pp. 73-91. http://geodesic.mathdoc.fr/item/DMGAA_2024_44_1_a4/