Related papers: Weakly Clean Ideal

As a generalization of nil clean ideal, we define weak nil clean ideal of a ring. An ideal $I$ of a ring $R$ is weak nil clean ideal if for any $x\in I$, either $x=e+n$ or $x=-e+n$, where $n$ is a nilpotent element and $e$ is an idempotent…

In this article, we introduce the concept of weakly $I$-clean ring, for any ideal $I$ of a ring $R$. We show that, for an ideal $I$ of a ring $R$, $R$ is uniquely weakly $I$-clean if and only if $R/I$ is semi boolean and idempotents can be…

Motivated by the concept of weakly clean rings, we introduce the concept of weakly $r$-clean rings. We define an element $x$ of a ring $R$ as weakly $r$-clean if it can be expressed as $x=r+e$ or $x=r-e$ where $e$ is an idempotent and $r$…

Motivated by the concept of clean index of rings, we introduce the concept of weak clean index of rings. For any element $a$ of a ring $R$ with unity, we define $ \chi(a)=\{e\in R\mid e^2=e\text{ and }a-e \mbox{ or } a+e \mbox{ is a…

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…

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 define the concepts of weakly precious and precious rings which generalize the notions of weakly clean and nil-clean rings. We obtain some fundamental properties of these rings. We also obtain certain subclasses of such rings and then…

Let $R$ be a commutative ring with non-zero identity. In this paper, we introduce the concept of weakly $J$-ideals as a new generalization of $J$-ideals. We call a proper ideal $I$ of a ring $R$ a weakly $J$-ideal if whenever $a,b\in R$…

In this paper, we introduce a new generalization of weakly prime ideals called $I$-prime. Suppose $R$ is a commutative ring with identity and $I$ a fixed ideal of $R$. A proper ideal $P$ of $R$ is $I$-prime if for $a, b \in R$ with $ab \in…

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if…

Let R be a commutative ring with $1\neq0$. In this paper, we introduce the concept of weakly 1-absorbing primary ideal which is a generalization of 1-absorbing ideal. A proper ideal $I$ of $R$ is called a weakly 1-absorbing primary ideal if…

In regard to our recent studies of rings with (strongly, weakly) nil-clean-like properties, we explore in-depth both the structural and characterization properties of those rings whose elements that are not units are weakly nil-clean. Group…

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…

{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…

This paper introduce and study weakly 1-absorbing prime ideals in commutative rings. Let $A$ be a commutative ring with a nonzero identity $1\neq 0$. A proper ideal $P$ of $A$ is said to be a weakly 1-absorbing prime ideal if for each…

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 define and study a particular case of von Neumann regular notion called a weak von Neumann regular ring. It shown that the polynomial ring $R[x]$ is weak von Neumann regular if and only if $R$ has exactly two idempotent…

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…

The goal of this article is to present the graded weakly $S$-primary ideals and $g$-weakly $S$-primary ideals which are extensions of graded weakly primary ideals. Let $R$ be a commutative graded ring, $S\subseteq h(R)$ and $P$ be a graded…

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…