Another Proof of Levitzki’s Theorem for Prime Rings and Application to k-skew Commuting Property

In this paper, we provide another proof of Levitzki’s theorem for prime rings with k!-torsion-free using the Vandermonde determinant, which states that a prime ring contains no nonzero nilpotent left or right ideal. As an application of this theorem, we prove that the k-skew commuting element in torsion-free prime *-rings with nonsymmetric element must be zero without using any deep structural results from ring theory.