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

Abstract

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.