Ideals and Symmetrc Left Bi-Derivations on Prime Rings

摘要

Let R be a non commutative 2, 3-torsion free prime ring and / 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 D_1(.,.):R × R→ R and D_2(.,.):R×R→R and B(.,.):R × R→R is a symmetric bi-additive mapping, such that (i) D_1(d_2(x),x) = 0, for all x ∈ I, (ii) d_1 [d_2(x)) = f(x), for all x ∈ I, where d_1 and d_2 are the traces of D_1 and D_2 respectively and f is trace of B, then either D_1 = 0 or D_2 = 0. If D acts as a left (resp. right) R-homomorphism on I, then D = 0.