# Article

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Keywords:
polynomial identity; nilpotent element; commutator ideal; associative ring; torsion free ring; center; commutativity
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 $[x,y] = 0$).
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