Short Communication On the Trace of a Permuting Tri-additive Mapping in Left sΓ-unital Γ-rings

Let M be 2 and 3 torsion-free left sΓ-unital Γ-rings. Let D: M ×M ×M → M be a permuting tri-additive mapping with the trace d(x) = D(x,x,x). Let σ: M → M be an endomorphism and τ: M → M an epimorphism. The objective of this paper is to prove the following: a) If d is (σ,τ)-skew commuting on M, then D = 0; b) If d is (τ,τ)-skew-centralizing on M, then d is (τ,τ)-commuting on M; c) If d is 2−(σ,τ)-commuting on M, then d is (σ,τ)-commuting on M