Action of n-derivations and n-multipliers on ideals of (semi)-prime rings

The present paper aims to investigate the containment of nonzero central ideal in a ring R when the trace of symmetric n-derivations satisfies some differential identities. Lastly, we prove that in a prime ring R of suitable torsion restriction, if D, G : Rn -> R are two nonzero symmetric n-derivations such that f(theta)theta+ theta g(theta) = 0 holds V theta E W, a nonzero left ideal of R where f and g are the traces of D and G, respectively, then either R is commutative or G acts as a left n-multiplier. Finally, we characterize symmetric n-derivations in terms of left n-multipliers.