Note on Lie ideals with symmetric bi-derivations in semiprime rings

Abstract

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