### A chain of Kurosh may have an arbitrary finite length

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We give some sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, normal EP element and strongly EP element by using solutions of certain equations.

We consider rings equipped with a closure operation defined in terms of a collection of commuting idempotents, generalising the idea of a topological closure operation defined on a ring of sets. We establish the basic properties of such rings, consider examples and construction methods, and then concentrate on rings which have a closure operation defined in terms of their lattice of central idempotents.

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) $[{x}^{m},{y}^{m}]=0$ for all $x,y\in R\setminus J\left(R\right)$ and (ii) $[{\left(xy\right)}^{m}+{y}^{m}{x}^{m},x]=0=[{\left(yx\right)}^{m}+{x}^{m}{y}^{m},x]$, 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{'}}$$...$

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 introduce related comparability for exchange ideals. Let $I$ be an exchange ideal of a ring $R$. If $I$ satisfies related comparability, then for any regular matrix $A\in {M}_{n}\left(I\right)$, there exist left invertible ${U}_{1},{U}_{2}\in {M}_{n}\left(R\right)$ and right invertible ${V}_{1},{V}_{2}\in {M}_{n}\left(R\right)$ such that ${U}_{1}{V}_{1}A{U}_{2}{V}_{2}=diag({e}_{1},\cdots ,{e}_{n})$ for idempotents ${e}_{1},\cdots ,{e}_{n}\in I$.

Let $R$ be an exchange ring in which all regular elements are one-sided unit-regular. Then every regular element in $R$ is the sum of an idempotent and a one-sided unit. Furthermore, we extend this result to exchange rings satisfying related comparability.

We characterize exchange rings having stable range one. An exchange ring $R$ has stable range one if and only if for any regular $a\in R$, there exist an $e\in E\left(R\right)$ and a $u\in U\left(R\right)$ such that $a=e+u$ and $aR\cap eR=0$ if and only if for any regular $a\in R$, there exist $e\in r.ann\left({a}^{+}\right)$ and $u\in U\left(R\right)$ such that $a=e+u$ if and only if for any $a,b\in R$, $R/aR\cong R/bR\u27f9aR\cong bR$.

It is shown that a ring $R$ is a $GM$-ring if and only if there exists a complete orthogonal set $\{{e}_{1},\cdots ,{e}_{n}\}$ of idempotents such that all ${e}_{i}R{e}_{i}$ are $GM$-rings. We also investigate $GM$-rings for Morita contexts, module extensions and power series rings.