Let $p$, $q$ and $r$ be fixed non-negative integers. In this note, it is shown that if $R$ is left (right) $s$-unital ring satisfying $[f\left(x^p{y}^q\right)-x^{r}y,x]=0$ ($[f\left(x^p{y}^q\right)-y{x}^r,x]=0$, respectively) where $f\left(\lambda \right)\in {\lambda }^2\mathbb{Z}\left[\lambda \right]$, then $R$ is commutative. Moreover, commutativity of $R$ is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.

In this paper we investigate commutativity of rings with unity satisfying any one of the properties: $$\begin{array}{cc}& \{1-g\left(y{x}^m\right)\}[y{x}^m-x^{r}f\left(y{x}^m\right)x^s,x]\{1-h\left(y{x}^m\right)\}=0,\hfill \\ & \{1-g\left(y{x}^m\right)\}[x^{m}y-x^{r}f\left(y{x}^m\right)x^s,x]\{1-h\left(y{x}^m\right)\}=0,\hfill \\ & y^t[x,y^n]=g\left(x\right)[f\left(x\right),y]h\left(x\right)\mathrm{a}nd[x,y^n]y^t=g\left(x\right)[f\left(x\right),y]h\left(x\right)\hfill \end{array}$$ for some $f\left(X\right)$ in $X^2\mathbb{Z}\left[X\right]$ and $g\left(X\right)$, $h\left(X\right)$ in $\mathbb{Z}\left[X\right]$, 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...