Polynomial identities that imply commutativity for rings Original Research Article

We deal with the polynomial identities of the form P=Q, where P and Q are monic monomials in two variables that have the same degree in each variable and they are different in the noncommutative and associative situation. (For example, x(xy)=(xy)x,x(xy2)=(xy)(yx) and so on.) We show the following two facts: If both P and Q have degree 3, then any 2-torsion free ring with identity that satisfies P=Q is commutative. While, if both P and Q have degree 4 and if the identity P=Q is not the type of xyyx=yxxy, then any 2,3-torsion free ring with identity that satisfies P=Q is commutative.