N-strong commutativity preserving maps of Г-rings

We introduce the notion of N-strong commutativity preserving Maps and N- anti- strong commutativity preserving Maps on a subset S of a Γ-ring R, in this paper, let N be a fixed positive integer we prove that if R is a semiprime Γ-ring with extended centroid CГ and additive map f from R onto R such that f either Γ- N- strong commutativity preserving Map or Γ-N-anti-strong commutativity preserving Map on R, then ⌈f(xn), x⌉a=0 for all x ∈ R, a ∈ Г.