Lie ideals, Jordan ideals and n-additive mappings in semiprime rings

Let [Formula: see text] be an [Formula: see text]-torsion free semiprime ring with center [Formula: see text] and [Formula: see text] be two [Formula: see text]-additive mappings with traces [Formula: see text], respectively. Sǒgǔtchǔ and Gǒlbasi [E. K. Sǒgǔtchǔ and Ǒ. Gǒlbasi, Commutativity theorems on Lie ideals with symmetric bi-derivations in semiprime rings, Asian Eur. J. Math. 16(7) (2023) 2350129] studied the following identities for symmetric bi-derivations: (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], (xii) [Formula: see text], for all [Formula: see text], where [Formula: see text] is a square closed Lie ideal of [Formula: see text] and then obtained that [Formula: see text] contains a nonzero central ideal. In this paper, we prove that the conclusion of above results holds for trace of any [Formula: see text]-additive mapping (not necessarily to be symmetric bi-derivation).