Related papers: On Zhou nil-clean rings
This paper investigates key properties of ZINC rings and their relationships with semicommutative and weakly semicommutative rings. We call an element $x$ of a ring $R$ zero insertive if $x=arb$ for some $a,b,r\in R$ such that $ab=0$ 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…
An element in a ring $R$ is called uniquely weakly nil-clean if every element in $R$ can be uniquely written as a sum or a difference of a nilpotent and an idempotent in the sense of very idempotents. The structure of the ring in which…
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 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 said to be clean if each element of $R$ can be written as the sum of a unit and an idempotent. $R$ is said to be weakly clean if each element of $R$ is either a sum or a difference of a unit and an idempotent, and $R$ is said…
A $*$-ring $R$ is called (strongly) $*$-clean if every element of $R$ is the sum of a projection and a unit (which commute with each other). In this note, some properties of $*$-clean rings are considered. In particular, a new class of…
Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The…
We introduce the class weakly nil clean rings, as rings R in which for every a\in R there exist an idempotent e and a nilpotent q such that a-e-q\in eRa. Every weakly nil clean ring is exchange. Weakly nil clean rings contain pi-regular…
In this paper we discuss several constructions that lead to new examples of nil-clean, clean, and exchange rings. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's…
We present new characterizations of the rings in which every element is the sum of two idempotents and a nilpotent that commute, and the rings in which every element is the sum of two tripotents and a nilpotent that commute. We prove that…
Let R be a ring (not necessarily commutative ring) with identity. The clean graph Cl(R) of a ring R is a graph with vertices in the form of ordered pair (e; u), where e is an idempotent of the ring R and u is a unit of the ring R. Two…
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 $u\in R$.…
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…
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…
A ring $R$ is a UU ring if every unit is unipotent, or equivalently if every unit is a sum of a nilpotent and an idempotent that commute. These rings have been investigated in C\u{a}lug\u{a}reanu \cite{C} and in Danchev and Lam \cite{DL}.…
The target of the present work is to give a new insight in the theory of {\it strongly weakly nil-clean} rings, recently defined by Kosan and Zhou in the Front. Math. China (2016) and further explored in detail by Chen-Sheibani in the J.…
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…
An element $a\in R$ is very clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and either $a-e$ or $a+e$ is invertible. A ring $R$ is very clean in case every element in $R$ is very clean. We explore the necessary and…