ON SYMMETRIC BIDERIVATIONS OF SEMIPRIME RINGS

Let R be a ring with centre Z(R). A biadditive symmetric mapping D(.,.) : R x R -> R is called symmetric biderivation if for any fixed y is an element of R, the mapping x bar right arrow D(x, y) is a derivation. A mapping f : R -> R defined by f (x) = D(x, x) is called the trace of D. In this paper we prove that a nonzero Lie ideal L of a semiprime ring R of characteristic different from two is central if it satisfies any one of the following properties: (1) f (xy) -/+ [x, y] is an element of Z(R), (ii) f (xy) -/+ [y, x] is an element of Z(R), (iii) f (xy) -/+ xy is an element of Z(R), (iv) f (xy) -/+ -/+ yx Z(R), (v) f([x,y] -/+ [x, y] is an element of Z(R), (vi) f ([x,y]) -/+ [y, x] is an element of Z(R), (vii) f ([x, y]) -/+ xy is an element of Z(R), (viii) f, ([x,y]) -/+ yx is an element of Z(R), (ix) f (xy) -/+ f (x) -/+ [x,y] is an element of Z(R), (x) f (xy) -/+ f (Y) -/+ [x,y] is an element of Z(R), (xi) f ([x,y] -/+ f (x) -/+ [y,x] is an element of Z(R), (xii) f ([x,y]) -/+ f(x) -/+ [y,x] is an element of Z(R), (xiii) f([x,y]) -/+ f(Y) -/+ [x,y] is an element of Z(R), (xiv) f ([x,y]) -/+ f(y) -/+ [y,x] is an element of Z(R), (xv) f([x,y]) -/+ f (xy) -/+ [x,y] is an element of Z(R), (xvi) f ([x, y]) -/+ f (xy) -/+ [y, x] is an element of Z(R), (xvii) f (x) f (y) -/+ [x, y] is an element of Z(R), (xviii) f (x) f (y) -/+ [y, x] is an element of Z(R), (xix) f (x) f (y)-/+ xy is an element of Z(R), (xx) f (x) f (y)-/+ yx is an element of Z(R), where f stands for the trace of a biadditive symmetric mapping D(.,.) : R x R -> R. Moreover, motivated by a well known theorem of Posner [11, Theorem 2] and a result of Deng and Bell [6, Theorem 2], we prove that if R admits a symmetric biderivation D such that the trace f of D is n-centralizing on L, then f is n-commuting on L.