Product of Traces of Permuting n-Derivations on Prime and Semiprime Ideals of a Ring

In this paper, we prove that if the product of the traces of two permuting n-derivations on a ring maps the ring into a prime ideal, then one of the traces maps the ring into the prime ideal. We will also extend the above result on the product of two or more traces under suitable characteristic restrictions. In fact, we proved that for a semiprime ideal \(\mathcal {P}\) of \(\mathcal {R}\) , \(d^r(\mathcal {P})\subseteq \mathcal {P}\) if and only if \(d(\mathcal {P})\subseteq \mathcal {P}\) for any positive integer r.