ON THE COMMUTATIVITY AND ANTICOMMUTATIVITY OF RINGS II

Abstract

It is shown that if $R$ is any associative ring such that for each $x,y\in R$, there exist an even natural number $m(x,y)$ and an odd natural number $n(x,y)$, depending on $x$ and $y$, with either $[x,y]^{m(x,y)} = [x,y]^{n(x,y)}$ or $(x\circ y)^{m(x, y)} = (x \circ y)^{n(x,y)}$, then either $[x,y]$ or $(x \circ y)$ is nilpotent for all $x, y$ in $R$. Moreover, $R$ is commutative if $R$ has no nonzero nil right ideals.