Some Commutativity Theorems for Non-Associative Rings

Abstract

In this section we prove some elementary results on commutativity of non-associative rings. In an elementary commutativity theorem for rings, Johnsen, Outcalt and Yaqub [1] showed that a non-associative ring R with unity satisfying $\left(xy\right)^{2} =x^{2} y^{2} $ for all x, y in R is necessarily commutative. Boers [2] extended this to show that such a ring is also associative provided it is 2 and 3-torsion free. Without using that the ring is 2 and 3-torsion free, we prove that any assosymmetric ring in which $\left(xy\right)^{2} =x^{2} y^{2} $is commutative and associative. Further, if Z(R) denotes the center of the ring R, we prove the commutativity of a 2-torsion free non-associative ring R satisfying any one of the following identities:\\ (1). $\left(xy\right)^{2} \in Z\left(R\right)$\\ (2). $\left(xy\right)^{2} -xy\in Z\left(R\right)$\\ (3). $\left(\left(xy\right)z\right)^{2} -\left(xy\right)z\in Z\left(R\right)$\\ (4). $\left[\left(xy\right)^{2} -yx,x\right]=0$or $\left[\left(xy\right)^{2} -yx,y\right]=0$\\ (5). $\left[x^{2} y^{2} -xy,x\right]=0$ or $\left[x^{2} y^{2} -xy,y\right]=0$\\ (6). $\left[\left(xy\right)^{2} -x^{2} y-xy^{2} +xy,x\right]=0$ or $\left[\left(xy\right)^{2} -x^{2} y-xy^{2} +xy,y\right]=0$ for all x, y, z in R.\\ At the end of this section we also give some example which show that the existence of the unity and 2-torsion free are essential in some results. We know that an assosymmetric ring R is a non-associtative ring in which $(x, y, z) = (P(x), P(y), P(z))$, where P is any permutation of x, y, z in R.