On commutativity of alternative rings with [xy^n x±yx^n y,x]=0

Let R be a n-torsion free with identity 1 ,In this article we investigate and prove the commutativity of alternative ring of the property(p_1 ), (p_2) and(p_3 ) under suitable constraints. (p_1 ) [xy^n x±yx^n y,x]=0 (p_2 ) [xy^n x±yx^n y,y]=0 (p_3) [x(〖xy)〗^2+(〖xy)〗^2 x,x] = 0. ∀ x,y ∈R.