# Article

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Keywords:
commutators; division rings; factorsubrings; polynomial identities; torsion-free rings
Summary:
In this paper we investigate commutativity of rings with unity satisfying any one of the properties: \begin{aligned} &\lbrace 1- g(yx^{m}) \rbrace \ [yx^{m} - x^{r} f (yx^{m}) \ x^s, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&\lbrace 1- g(yx^{m}) \rbrace \ [x^{m} y - x^{r} f (yx^{m}) x^{s}, x] \lbrace 1- h (yx^{m}) \rbrace = 0, \\&y^{t} [x,y^{n}] = g (x) [f (x), y] h (x)\ {\mathrm and} \ \ [x,y^{n}] \ y^{t} = g (x) [f (x), y] h (x) \end{aligned} for some $f(X)$ in $X^{2} {\mathbb Z}[X]$ and $g(X)$, $h(X)$ in ${\mathbb Z} [X]$, where $m \ge 0$, $r \ge 0$, $s \ge 0$, $n > 0$, $t > 0$ are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements $x$ and $y$ for their values. Further, under different appropriate constraints on commutators, commutativity of rings has been studied. These results generalize a number of commutativity theorems established recently.
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