Commutativity theorems on Lie ideals with symmetric bi-derivations in semiprime rings

Abstract

Let [Formula: see text] be a semiprime ring, [Formula: see text] a square-closed Lie ideal of [Formula: see text] and [Formula: see text] a symmetric bi-derivation and [Formula: see text] be the trace of [Formula: see text] In this paper, we shall prove that [Formula: see text] contains a nonzero central ideal if any one of the following holds: (i) [Formula: see text] (ii) [Formula: see text] (iii) [Formula: see text] (iv) [Formula: see text] (v) [Formula: see text] (vi) [Formula: see text] (vii) [Formula: see text] (viii) [Formula: see text] (ix) [Formula: see text] (x) [Formula: see text] (xi) [Formula: see text] and (xii) [Formula: see text], for all [Formula: see text]