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### On structure of certain periodic rings and near-rings.

International Journal of Mathematics and Mathematical Sciences

### Commutativity of rings with polynomial constraints

Czechoslovak Mathematical Journal

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 $\left[f\left({x}^{p}{y}^{q}\right)-{x}^{r}y,x\right]=0$ ($\left[f\left({x}^{p}{y}^{q}\right)-y{x}^{r},x\right]=0$, respectively) where $f\left(\lambda \right)\in {\lambda }^{2}ℤ\left[\lambda \right]$, 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.

### Commutativity of rings with constraints involving a subset

Czechoslovak Mathematical Journal

Suppose that $R$ is an associative ring with identity $1$, $J\left(R\right)$ the Jacobson radical of $R$, and $N\left(R\right)$ the set of nilpotent elements of $R$. Let $m\ge 1$ be a fixed positive integer and $R$ an $m$-torsion-free ring with identity $1$. The main result of the present paper asserts that $R$ is commutative if $R$ satisfies both the conditions (i) $\left[{x}^{m},{y}^{m}\right]=0$ for all $x,y\in R\setminus J\left(R\right)$ and (ii) $\left[{\left(xy\right)}^{m}+{y}^{m}{x}^{m},x\right]=0=\left[{\left(yx\right)}^{m}+{x}^{m}{y}^{m},x\right]$, for all $x,y\in R\setminus J\left(R\right)$. This result is also valid if (i) and (ii) are replaced by (i)${}^{\text{'}}$ $\left[{x}^{m},{y}^{m}\right]=0$ for all $x,y\in R\setminus N\left(R\right)$ and (ii)${}^{\text{'}}$ $\left[{\left(xy\right)}^{m}+{y}^{m}{x}^{m},x\right]=0=\left[{\left(yx\right)}^{m}+{x}^{m}{y}^{m},x\right]$ for all $x,y\in R\setminus N\left(R\right)$. Other similar commutativity...

### Commutativity of rings through a Streb’s result

Czechoslovak Mathematical Journal

In this paper we investigate commutativity of rings with unity satisfying any one of the properties: $\begin{array}{cc}& \left\{1-g\left(y{x}^{m}\right)\right\}\phantom{\rule{4pt}{0ex}}\left[y{x}^{m}-{x}^{r}f\left(y{x}^{m}\right)\phantom{\rule{4pt}{0ex}}{x}^{s},x\right]\left\{1-h\left(y{x}^{m}\right)\right\}=0,\hfill \\ & \left\{1-g\left(y{x}^{m}\right)\right\}\phantom{\rule{4pt}{0ex}}\left[{x}^{m}y-{x}^{r}f\left(y{x}^{m}\right){x}^{s},x\right]\left\{1-h\left(y{x}^{m}\right)\right\}=0,\hfill \\ & {y}^{t}\left[x,{y}^{n}\right]=g\left(x\right)\left[f\left(x\right),y\right]h\left(x\right)\phantom{\rule{4pt}{0ex}}\mathrm{a}nd\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\left[x,{y}^{n}\right]\phantom{\rule{4pt}{0ex}}{y}^{t}=g\left(x\right)\left[f\left(x\right),y\right]h\left(x\right)\hfill \end{array}$ for some $f\left(X\right)$ in ${X}^{2}ℤ\left[X\right]$ and $g\left(X\right)$, $h\left(X\right)$ in $ℤ\left[X\right]$, 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 generalize...

### Classification of rings satisfying some constraints on subsets

Archivum Mathematicum

Let $R$ be an associative ring with identity $1$ and $J\left(R\right)$ the Jacobson radical of $R$. Suppose that $m\ge 1$ is a fixed positive integer and $R$ an $m$-torsion-free ring with $1$. In the present paper, it is shown that $R$ is commutative if $R$ satisfies both the conditions (i) $\left[{x}^{m},{y}^{m}\right]=0$ for all $x,y\in R\setminus J\left(R\right)$ and (ii) $\left[x,\left[x,{y}^{m}\right]\right]=0$, for all $x,y\in R\setminus J\left(R\right)$. This result is also valid if (ii) is replaced by (ii)’ $\left[{\left(yx\right)}^{m}{x}^{m}-{x}^{m}{\left(xy\right)}^{m},x\right]=0$, for all $x,y\in R\setminus N\left(R\right)$. Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).

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