Related papers: r-clean rings

A ring $R$ is said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring $\mathbb…

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 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…

We consider in-depth and characterize in certain aspects those rings whose non-units are strongly nil-clean in the sense that they are a sum of commuting nilpotent and idempotent. In addition, we examine those rings in which the non-units…

A unital ring is called clean (resp. strongly clean) if every element can be written as the sum of an invertible element and an idempotent (resp. an invertible element and an idempotent that commutes). T.Y. Lam proposed a question: which…

We consider and study those rings in which each nil-clean or clean element is uniquely nil-clean. We establish that, for abelian rings, these rings have a satisfactory description and even it is shown that the classes of abelian rings and…

{Generalizing the notion of nil cleanness from \cite{D13}, in parallel to \cite{DM14}, we define the concept of {\it weak nil cleanness} for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition…

We construct an example of a unit-regular ring which is not strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between…

A ring $R$ is nil-clean if every element in $R$ is the sum of an idempotent and a nilpotent. A ring $R$ is abelian if every idempotent is central. We prove that if $R$ is abelian then $M_n(R)$ is nil-clean if and only if $R/J(R)$ is Boolean…

We define and consider in-depth the so-called $C\Delta$ rings as those rings $R$ whose elements are a sum of an element in $C(R)$ and of an element in $\Delta(R)$. Our achieved results somewhat strengthen these recently obtained by…

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 periodic provided that for any $a\ in R$ there exist distinct elements $m,n \in {\Bbb N}$ such that $a^m=a^n$. We shall prove that periodicity is inherited by a type of generalized matrix rings.We define strongly periodic…

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…

Many authors have investigated the behavior of strong cleanness under certain ring extensions. In this note, we prove that if $R$ is a ring which is complete with respect to an ideal $I$ and if $x$ is an element of $R$ whose image in $R/I$…

A ring R is a Zhou nil-clean ring if every element in R is the sum of two tripotents and a nilpotent that commute. In this paper, Zhou nil-clean rings are further discussed with an emphasis on their relations with polynomials, idempotents…

A ring is called clean if every element is the sum of an invertible element and an idempotent. This paper investigates the cleanness of AW*-algebras. We prove that all finite AW*-algebras are clean, affirmatively solving a question posed by…

Let $R$ be an associative ring with identity, $C(R)$ denote the center of $R$, and $g(x)$ be a polynomial in the polynomial ring $C(R)[x]$. $R$ is called strongly $g(x)$-clean if every element $r \in R$ can be written as $r=s+u$ with…

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…

A ring with an involution * is called strongly $J$-*-clean if every element is a sum of a projection and an element of the Jacobson radical that commute. In this article, we prove several results characterizing this class of rings. It is…

A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every…