A commutativity theorem for associative rings

summary:Let $m > 1, s\geq 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p = p(x) \geq 0, q = q(x) \geq 0, n = n(x) \geq 0, r = r(x) \geq 0 $ such that either $ x^{p}[x^{n},y]x^{q} = x^{r}[x,y^{m}]y^{s} $ or $ x^{p}[x^{n},y]x^{q} = y^{s}[x,y^{m}]x^{r} $ for all $ y \in R $. In the present paper it is shown that $R$ is commutative if it satisfies the property $Q(m)$ (i.e. for all $x,y \in R, m[x,y] = 0$ implies $[x,y] = 0$)