### A combinatorial commutativity property for rings.

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Let $m>1,s\ge 1$ be fixed positive integers, and let $R$ be a ring with unity $1$ in which for every $x$ in $R$ there exist integers $p=p\left(x\right)\ge 0,q=q\left(x\right)\ge 0,n=n\left(x\right)\ge 0,r=r\left(x\right)\ge 0$ 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 $Q\left(m\right)$ (i.e. for all $x,y\in R,m[x,y]=0$ implies $[x,y]=0$).

In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let ${J}^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very ${J}^{\#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of ${J}^{\#}$. A ring $R$ is said to be very ${J}^{\#}$-clean in case every element in $R$ is very ${J}^{\#}$-clean. We prove that every very ${J}^{\#}$-clean ring is strongly $\pi $-rad clean and has stable range one. It is shown...

A ring $R$ is defined to be left almost Abelian if $ae=0$ implies $aRe=0$ for $a\in N\left(R\right)$ and $e\in E\left(R\right)$, where $E\left(R\right)$ and $N\left(R\right)$ stand respectively for the set of idempotents and the set of nilpotents of $R$. Some characterizations and properties of such rings are included. It follows that if $R$ is a left almost Abelian ring, then $R$ is $\pi $-regular if and only if $N\left(R\right)$ is an ideal of $R$ and $R/N\left(R\right)$ is regular. Moreover it is proved that (1) $R$ is an Abelian ring if and only if $R$ is a left almost Abelian left idempotent reflexive ring. (2) $R$ is strongly...

Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = ${\sum}_{i\ge 0}{r}_{i}{x}^{i}\in R\left[\left[x\right]\right]$ : ∃ 0 ≤ n∈ ℤ such that ${r}_{i}\in I$, ∀ i ≥ n. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with a certain ring...

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) $[{x}^{m},{y}^{m}]=0$ for all $x,y\in R\setminus J\left(R\right)$ and (ii) $[x,[x,{y}^{m}]]=0$, for all $x,y\in R\setminus J\left(R\right)$. This result is also valid if (ii) is replaced by (ii)’ $[{\left(yx\right)}^{m}{x}^{m}-{x}^{m}{\left(xy\right)}^{m},x]=0$, 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]).