Related papers: Very clean matrices over local rings
An element $a$ of a ring $R$ is called perfectly clean if there exists an idempotent $e\in comm^2(a)$ such that $a-e\in U(R)$. A ring $R$ is perfectly clean in case every element in $R$ is perfectly clean. In this paper, we investigate…
An element $a\in R$ is provided that there exists an idempotent $e\in R$ such that $a-e\in U(R), ae=ea$ and $eae\in J(eRe)$. In this article, we investigate strongly rad-clean matrices over a commutative local ring. We completely determine…
A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute with each other. A recent result of Borooah, Diesl and Dorsey \cite{BDD05a} completely characterized the commutative local rings…
An element of a ring $R$ is strongly $P$-clean provided that it can be written as the sum of an idempotent and a strongly nilpotent element that commute. A ring $R$ is strongly $P$-clean in case each of its elements is strongly $P$-clean.…
A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we…
A ring $R$ is uniquely (strongly) clean provided that for any $a\in R$ there exists a unique idempotent $e\in R \big(\in comm(a)\big)$ such that $a-e\in U(R)$. Let $R$ be a uniquely bleached ring. We prove, in this note, that $R$ is…
An element $a$ in a ring $R$ is strongly J-clean if it is the sum of an idempotent and an element in the Jacobson radical that commutes. We characterize the strongly J-clean $2\times 2$ matrices over 2-projective-free non-commutative rings.
A ring $R$ is called strongly clean if every element of $R$ is the sum of a unit and an idempotent that commute. By {\rm SRC} factorization, Borooah, Diesl, and Dorsey \cite{BDD051} completely determined when ${\mathbb M}_n(R)$ over a…
An element $a$ of a ring $R$ is called \emph{strongly $J$-clean} provided that there exists an idempotent $e\in R$ such that $a-e\in J(R)$ and $ae=ea$. A ring $R$ is \emph{strongly $J$-clean} in case every element in $R$ is strongly…
A ring $R$ is strongly clean provided that every element in $R$ is the sum of an idempotent and a unit that commutate. Let $T_n(R,\sigma)$ be the skew triangular matrix ring over a local ring $R$ where $\sigma$ is an endomorphism of $R$. We…
A ring $R$ is trinil clean if every element in $R$ is the sum of a tripotent and a nilpotent. If $R$ is a 2-primal strongly 2-nil-clean ring, we prove that $M_n(R)$ is trinil clean for all $n\in {\Bbb N}$. Furthermore, we show that the…
Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $\mathbb{M}_n(R)$ is strongly clean for…
A commutative ring $R$ is projective free provided that every finitely generated $R$-module is free. An element in a ring is strongly clean provided that it is the sum of an idempotent and a unit that commutates. Let $R$ be a…
An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices…
A ring $R$ is said to be $n$-clean if every element can be written as a sum of an idempotent and $n$ units. The class of these rings contains clean ring and $n$-good rings in which each element is a sum of $n$ units. In this paper, we show…
The literature about strongly clean matrices over commutative rings is quite extensive. The sharpest results are about matrices over commutative local rings, for example those by Borooah, Diesl and Dorsey. The purpose of this note is to…
A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Let R be a Zhou nil-clean ring. If R is 2-primal (of bounded index), we prove that every square matrix over R is the sum of two…
A ring $R$ is feckly clean provided that for any $a\in R$ there exists an element $e\in R$ and a full element $u\in R$ such that $a=e+u, eR(1-e)\subseteq J(R)$. We prove that a ring $R$ is feckly clean if and only if for any $a\in R$, there…
An element of a ring is unique clean if it can be uniquely written as the sum of an idempotent and a unit. A ring $R$ is uniquely $\pi$-clean if some power of every element in $R$ is uniquely clean. In this article, we prove that a ring $R$…
A ring R is a strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. In this paper, strongly…