Ideals and Symmetrc Left Bi-Derivations on Prime Rings

Let 𝑅 be a non commutative 2, 3-torsion free prime ring and 𝐼 be a non zero ideal of 𝑅. Let 𝐷 (.,.) :𝑅×𝑅→𝑅 be a symmetric left bi-derivation such that 𝐷(𝐼,𝐼)⊂𝐼 and 𝑑 is a trace of 𝐷. If (i) [𝑑(𝑥),𝑥] =0, for all 𝑥∈𝐼, (ii) [𝑑(𝑥),𝑥] ∈𝑍(𝑅), for all 𝑥∈𝐼, then 𝐷=0. Suppose that there exists symmetric left bi-derivations 𝐷₁ (.,.) :𝑅×𝑅→𝑅 and 𝐷₂ (.,.) :𝑅×𝑅→𝑅 and 𝐵 (.,.) :𝑅×𝑅→𝑅 is a symmetric bi-additive mapping, such that (i) 𝐷₁ (𝑑₂ (𝑥) ,𝑥) =0, for all 𝑥∈𝐼, (ii) 𝑑₁ (𝑑₂ (𝑥)) =𝑓(𝑥), for all 𝑥∈𝐼, where 𝑑₁ and 𝑑₂ are the traces of 𝐷₁ and 𝐷₂ respectively and 𝑓 is trace of 𝐵, then either 𝐷₁=0 or 𝐷₂=0. If 𝐷 acts as a left (resp. right) 𝑅-homomorphism on 𝐼, then 𝐷=0.