Polynomial identities that imply commutativity for rings

We deal with the polynomial identities of the form P 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 (xy(2)) = (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. (C) 2002 Elsevier Science Inc. All rights reserved.