ON PRIME AND SEMIPRIME RINGS WITH SYMMETRIC n-DERIVATIONS

Abstract

Let n ≥ 2 be a fixed positive integer and let R be a noncommutative n!-torsion free semiprime ring. Suppose that there exists a symmetric n-derivation ∆ : Rn → R such that the trace of ∆ is centralizing on R. Then the trace is commuting on R. If R is a n!-torsion free prime ring and ∆ (cid:54)= 0 under the same condition. Then R is commutative.

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