Related papers: Rings over which every matrix is the sum of a trip…

Motivated by the concept of clean ideals, we introduce the notion of nil clean ideals of a ring. We define an ideal $I$ of a ring $R$ to be nil clean ideal if every element of $I$ can be written as a sum of an idempotent and a nilpotent…

An element $x \in R$ is considered (strongly) nil-clean if it can be expressed as the sum of an idempotent $e \in R$ and a nilpotent $b \in R$ (where $eb = be$). If for any $x \in R$, there exists a unit $u \in R$ such that $ux$ is…

A *-ring $R$ is called a strongly nil-*-clean ring if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this article, we show that $R$ is a strongly nil-*-clean ring if and only if…

We study those rings in which all invertible elements are weakly nil-clean calling them {\it UWNC rings}. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are…

We continue the study in-depth of the so-called $n$-UU rings for any $n\geq 1$, that were defined by the first-named author in Toyama Math. J. (2017) as those rings $R$ for which $u^n-1$ is always a nilpotent for every unit $u\in R$.…

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…

Question 3 of [3] asks whether the matrix ring Mn(R) is nil clean, for any nil clean ring R. It is shown that positive answer to this question is equivalent to positive solution for Kothe's problem in the class of algebras over the field…

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.

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…

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…

We prove that if an involution in a ring is the sum of an idempotent and a nilpotent then the idempotent in this decomposition must be 1. As a consequence, we completely characterize weakly nil-clean rings introduced recently in [Breaz,…

For an element $a \in R$, let $\eta(a)=\{e\in R\mid e^2=e\mbox{ and }a-e\in \mbox{nil}(R)\}$. The nil clean index of $R$, denoted by NinA$(R)$, is defined as Nin$(R)=\sup \{\mid \eta(a)\mid: a\in R\}$. In this article we have characterized…

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 $R$…

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…

We significantly strengthen results on the structure of matrix rings over finite fields and apply them to describe the structure of the so-called weakly $n$-torsion clean rings. Specifically, we establish that, for any field $F$ with either…

In this paper, strongly clean ring defined by W. K. Nicholson in 1999 has been generalized to n-strongly clean, {\Sigma}-strongly clean and with the help of example it has been shown that there exists a ring, which is n-strongly clean and…

We introduce the concept of a weak nil clean ring, a generalization of nil clean ring, which is nothing but a ring with unity in which every element can be expressed as sum or difference of a nilpotent and an idempotent. Further if the…

We systematically study those rings whose non-units are a sum of an idempotent and a nilpotent. Some crucial characteristic properties are completely described as well as some structural results for this class of rings are obtained. This…

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…

In this paper we introduce and study the notion of a graded (strongly) nil clean ring which is group graded. We also deal with extensions of graded (strongly) nil clean rings to graded matrix rings and to graded group rings. The question of…