### Elements in exchange rings with related comparability.

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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.

A matrix $A\in {M}_{n}\left(R\right)$ is $e$-clean provided there exists an idempotent $E\in {M}_{n}\left(R\right)$ such that $A-E\in {\mathrm{GL}}_{n}\left(R\right)$ and $detE=e$. We get a general criterion of $e$-cleanness for the matrix $\left[[{a}_{1},{a}_{2},\cdots ,{a}_{n+1}]\right]$. Under the $n$-stable range condition, it is shown that $\left[[{a}_{1},{a}_{2},\cdots ,{a}_{n+1}]\right]$ is $0$-clean iff $({a}_{1},{a}_{2},\cdots ,{a}_{n+1})=1$. As an application, we prove that the $0$-cleanness and unit-regularity for such $n\times n$ matrix over a Dedekind domain coincide for all $n\ge 3$. The analogous for $(s,2)$ property is also obtained.

An exchange ring $R$ is strongly separative provided that for all finitely generated projective right $R$-modules $A$ and $B$, $A\oplus A\cong A\oplus B\Rightarrow A\cong B$. We prove that an exchange ring $R$ is strongly separative if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exist $u,v\in S$ such that $au=bv$ and $Su+Sv=S$ if and only if for any corner $S$ of $R$, $aS+bS=S$ implies that there exists a right invertible matrix $\left(\begin{array}{cc}a& b\\ & *\end{array}\right)\in {M}_{2}\left(S\right)$. The dual assertions are also proved.

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 $a,b\in R$, $R/aR\cong R/bR\u27f9aR\cong bR$.

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 ${e}_{1},\cdots ,{e}_{n}\in I$.

In this paper, we prove that unit ideal-stable range condition is right and left symmetric.

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.

In this paper we investigate the related comparability over exchange rings. It is shown that an exchange ring R satisfies the related comparability if and only if for any regular x C R, there exists a related unit w C R and a group G in R such that wx C G.

A $*$-ring $R$ is strongly 2-nil-$*$-clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$-rings are obtained. We prove that a $*$-ring $R$ is strongly 2-nil-$*$-clean if and only if for all $a\in R$, ${a}^{2}\in R$ is strongly nil-$*$-clean, if and only if for any $a\in R$ there exists a $*$-tripotent $e\in R$ such that $a-e\in R$ is nilpotent and $ea=ae$, if and only if $R$ is a strongly $*$-clean SN ring, if and only if $R$ is abelian, $J\left(R\right)$ is nil and $R/J\left(R\right)$ is $*$-tripotent. Furthermore, we explore the structure...

We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; ${\mathbb{Z}}_{3}\oplus {\mathbb{Z}}_{3}$; ${\mathbb{Z}}_{3}\oplus B$ where $B$ is a Boolean ring; local ring with nil Jacobson radical; ${M}_{2}\left({\mathbb{Z}}_{2}\right)$ or ${M}_{2}\left({\mathbb{Z}}_{3}\right)$; or the ring of a Morita context with zero pairings where the underlying rings are ${\mathbb{Z}}_{2}$ or ${\mathbb{Z}}_{3}$.

A ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $R$ be abelian, and let $n\in \mathbb{N}$. We prove that ${M}_{n}\left(R\right)$ is nil clean if and only if $R/J\left(R\right)$ is Boolean and ${M}_{n}\left(J\left(R\right)\right)$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R/J\left(R\right)$ is ${\mathbb{Z}}_{3}$, $B$ or ${\mathbb{Z}}_{3}\oplus B$ where $B$ is a Boolean ring, and that ${M}_{n}\left(R\right)$ is weakly nil clean if and only if ${M}_{n}\left(R\right)$ is nil clean for all $n\ge 2$.

We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a $2\times 2$ matrix $A$ over a projective-free ring $R$ is strongly $J$-clean if and only if $A\in J\left({M}_{2}\left(R\right)\right)$, or ${I}_{2}-A\in J\left({M}_{2}\left(R\right)\right)$, or $A$ is similar to $\left({\textstyle \begin{array}{cc}0& \lambda \\ 1& \mu \end{array}}\right)$, where $\lambda \in J\left(R\right)$, $\mu \in 1+J\left(R\right)$, and the equation ${x}^{2}-x\mu -\lambda =0$ has a root in $J\left(R\right)$ and a root in $1+J\left(R\right)$. We further prove that $f\left(x\right)\in R\left[\right[x\left]\right]$ is strongly $J$-clean if $f\left(0\right)\in R$ be optimally $J$-clean.

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