Advanced Search

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 $n\ge 2$.

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 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 $\left(\begin{array}{cc}0& \lambda \\ 1& \mu \end{array}\right)$, 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.

A ring $R$ is feebly nil-clean if for any $a\in R$ there exist two orthogonal idempotents $e,f\in R$ and a nilpotent $w\in R$ such that $a=e-f+w$. Let $R$ be a 2-primal feebly nil-clean ring. We prove that every matrix ring over $R$ is feebly nil-clean. The result for rings of bounded index is also obtained. These provide many classes of rings over which every matrix is the sum of orthogonal idempotent and nilpotent matrices.

We present new characterizations of the rings for which every element is a sum of two tripotents and a nilpotent that commute. These extend the results of Z. L. Ying, M. T. Koşan, Y. Zhou (2016) and Y. Zhou (2018).