Bi-derivations on Semiprime Rings

The main purpose of this paper is studying and investigating some results concerning a symmetric bi-derivation D:R×R→R and d the trace of D ,on prime rings and semiprime rings R, where R admits for a symmetric bi-derivation D satisfying some conditions on R, (i). ([d(x),x]) n =0 for all x ∈R. (ii). D 1 (d 2 (x),x) n =0 holds for all x ∈R. (iii). (d 1 (d 2 (x))- f(x)) n =0 holds for all x ∈R. Where n be a positive integer and R be a 2-torsion and 3-torsion free.