Research Journal of Science and Technology

Let R be a non commutative 2, 3-torsion free prime ring and I be a non zero ideal of R. Let Dd(.,.):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,ε I (ii) [d(x),x] ε Z(Rx), for all xε I then D =0. Suppose that there exists symmetric left bi-derivations D₁d(.,.)R×R → R and D₂d(.,.)R×R → R and Bd(.,.)R×R → R is a symmetric bi-additive mapping, such that (i) D₁ (d₂;d(x),x) =0, for all x ε I (ii) d₁ (d₂;d(x),) = f(x), for all x ε 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.