SOME RESULT ON LIE IDEALS WITH SYMMETRIC REVERSE BI-DERIVATIONS IN SEMIPRIME RINGS I

Let R be a semiprime ring, U a square-closed Lie ideal of R and D : R x R -> R a symmetric reverse bi-derivation and d be the trace of D: In the present paper, we shall prove that R is commutative ring if any one of the following holds: i) d(U) = (0); ii)d(U) subset of Z; iii)[d (x), y] is an element of Z, iv)d(x)oy is an element of Z, v)d ([x, y]) +/- [d(x), y] is an element of Z, vi)d (x o y) +/- (d(x) oy) is an element of Z, vii)d ([x, y]) +/- d(x) oy is an element of Z viii)d (x o y) +/- [d(x), y] is an element of Z, ix)d(x) oy +/-[d(y), x] is an element of Z, x)d([x, y]) (d(x) oy) [d(y), x] is an element of Z xi)[d(x), y]+/- [g(y), x] is an element of Z, for all x, y is an element of U, where G : R x R -> R is symmetric reverse bi-derivation such that g is the trace of G.z