Ajay Sharma, Dhiren Kumar Basnet
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 element of $R$. In this article we discuss various properties of nil clean ideals.
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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 idempotent and nilpotent commute the ring is called weak* nil clean. We characterize all $n\in \mathbb{N}$, for which $\mathbb{Z}_n$ is weak nil clean but not nil clean. We show that if $R$ is a weak* nil clean and $e$ is an idempotent in $R$, then the corner ring $eRe$ is also weak* nil clean. Also we discuss $S$-weak nil clean rings and their properties, where…
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 F_2. Other equivalent problems are also discussed. The classes of conjugate clean and conjugate nil clean rings, which lie strictly between uniquely (nil) clean and (nil) clean rings are introduced and investigated.
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 element of $R$. Some interesting properties of weak nil clean ideal and its relation with weak nil clean ring have been discussed.
We describe nil-clean companion matrices over fields.
Rings whose Nil-Clean and Clean Elements are Uniquely Nil-Clean
研究每个 nil-clean 或 clean 元素均为 uniquely nil-clean 的环，证明了 abelian rings 与此类环的类重合，并对 semipotent 情形给出了完全刻画。
In this article, we introduce a new graph theoretic structure associated with a finite commutative ring, called nil clean divisor graph. For a ring $R$, nil clean divisor graph is denoted by $G_N(R)$, where the vertex set is $\{x\in R\,:\, x\neq 0, \,\exists\, y(\neq 0, \neq x)\in R$ such that $xy$ is nil clean$\}$, two vertices $x$ and $y$ are adjacent if $xy$ is a nil clean element. We prove some interesting results of nil clean divisor graph of a ring.
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 every idempotent in $R$ is a projection, $R$ is periodic, and $R/J(R)$ is Boolean. For any commutative *-ring $R$, we prove that the algebraic extension $R[i]$ where $i^2=μi+η$ for some $μ,η\in R$ is strongly nil-*-clean if and only if $R$ is strongly nil-*-clean and $μη$ is nilpotent. The relationships between Boolean *-rings and strongly nil-*-clean rings are …
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 properties (the infinitesimal deformation case). From infinitesimal deformations we go to full deformations and prove that any formal deformation of a clean (exchange) ring is itself clean (exchange). Examples of nil-clean, clean, and exchange rings arising from poset algebras are also discussed.
A ring R is Yaqub nil-clean if for any a\in R, a-a^3 or a+a^3 is nilpotent for all a\in R. We prove that a ring R is Yaqub nil-clean if and only if for any a\in R, there exists some e^3=e\in R, such that a-e or a+3e is nilpotent and ae=ea.
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 when nil cleanness of the component, which corresponds to the neutral element of a group, implies graded nil cleanness of the whole graded ring is examined. Similar question is discussed in the case of groupoid graded rings as well.
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 tripotents and a nilpotent. These provides a large class of rings over which every square matrix has such decompositions by tripotent and nilpotent matrices.
In this article, we have defined nil clean graph of a ring $R$. The vertex set is the ring $R$, two ring elements $a$ and $b$ are adjacent if and only if $a + b$ is nil clean in $R$. Graph theoretic properties like girth, dominating set, diameter etc. of nil clean graph have been studied for finite commutative rings.
In this article, the concept of nil clean graph of a ring has been generalised to weakly nil clean graph of a ring and graph theoretic properties like girth, clique number, diameter and chromatic index of the graph have been studied for a finite commutative ring.
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 and $M_n(J(R))$ is nil. This extend the main results of Breaz et al. ~\cite{BGDT} and that of Koşan et al.~\cite{KLZ}.
The notion of clean rings and 2-good rings have many variations, and have been widely studied. We provide a few results about two new variations of these concepts and discuss the theory that ties these variations to objects and properties of interest to noncommutative algebraists. A ring is called nil-good if each element in the ring is the sum of a nilpotent element and either a unit or zero. We establish that the ring of endomorphisms of a module over a division is nil-good, as well as some basic consequences. We then define a new property we call nil-good clean, the condition that an elemen…
For an element $a \in R$, let $η(a)=\{e\in R\mid e^2=e\mbox{ and }a-e\in \mbox{nil}(R)\}$. The nil clean index of $R$, denoted by NinA$(R)$, is defined as Nin$(R)=\sup \{\mid η(a)\mid: a\in R\}$. In this article we have characterized formal triangular matrix ring $\begin{pmatrix}A & M0 & B\end{pmatrix}$ with nil clean index $4$.
Graded m-nil clean ring
本文引入分次 m-nil 清洁环的概念，推广了分次 nil-清洁环，并研究了其基本性质、分次群环、分次矩阵环和分次融合代数的继承条件。
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 every zero-divisor is uniquely weakly nil-clean is completely determined. We prove that every zero-divisor in a ring $R$ is uniquely weakly nil-clean if and only if $R$ is a D-ring, or $R$ is abelian, periodic, and $R/J(R)$ is isomorphic to a field $F$, ${\Bbb Z}_{3}\oplus {\Bbb Z}_{3}$, ${\Bbb Z}_{3}\oplus B$ where $B$ is Boolean, or a Boolean ring. As a specif…
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, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl., DOI: 10.1142/S0219498816501486].
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 2-nil-clean rings are studied with an emphasis on their relations with feebly clean rings. This work shows new interesting connections between strongly 2-nil-clean rings and weakly exchange rings
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 show that $M_n(R)$ is 2-nil-clean for all $n\in {\Bbb N}$. We also prove that the matrix ring is 2-nil-clean for a strongly 2-nil-clean ring of bounded index. These provide many classes of rings over which every matrix is the sum of two idempotents and a nilpotent.
{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 theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is {\it weakly nil-clean}.} It is also completely determined when a matrix ring over a division ring is weakly nil-clean.
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 offer new kinds of weakly clean rings and nil-clean rings.
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 matrix ring over a strongly 2-nil-clean ring of bounded index is trinil clean. We thereby provide various type of rings over which every matrix is the sum of a tripotent and a nilpotent.
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 rings as a proper subclass, and these two classes coincide in the case of central idempotents. Every weakly nil clean ring of bounded index and every weakly nil clean PI-ring is strongly pi-regular. The center of a weakly nil clean ring is strongly pi-regular, and consequently, every weakly nil clean ring is a corner of a clean ring. These results extend Azumay…
On Quasi-Nil Clean Rings
本文引入拟幂零清洁环的概念，其中每个元素可表示为拟幂等元与幂零元之和，并研究其结构性质、与现有环类的关系及交换环的拟布尔商刻画。
On some extensions of strongly unit nil-clean rings
本文推广了强单位幂零清洁环的概念，给出了群环单位正则性的刻画条件，完全描述了$\mathbb{Z}_nG$的单位正则性，并探讨了Morita上下文、矩阵环和群环中的强单位幂零清洁性。
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 and 2- idempotents. any a \in R, there exists e 2 Z[a] such that a-e \in R is nilpotent and e^5 = 5e^3 -4e, if and only if for any a 2 R, there exist idempotents e; f; g; h 2 Z[a] and a nilpotent w such that a = e+f +g+h+w, if and only if for any a 2 R, there exist 2-idempotents e; f 2 Z[a] and a nilpotent w 2 R such that a = e + f + w, if and only if for any…
Rings Whose Clean and Nil-Clean Elements Have Some Clean-Like Properties
本文定义了CSNC和NCUC两类环，分别要求清洁元素为强幂零清洁、幂零清洁元素为唯一清洁，并扩展了Calugareanu-Zhou和Cui-Danchev-Jin的相关研究成果。
We characterize the nil clean matrix rings over fields. As a by product, it is proved that the full matrix rings with coefficients in commutative nil-clean rings are nil-clean, and we obtain a complete characterization of the finite rank Abelian groups with nil clean endomorphism ring and the Abelian groups with strongly nil clean endomorphism ring, respectively.
We prove that a Lie nilpotent one-sided ideal of an associative ring $R$ is contained in a Lie solvable two-sided ideal of $R$. An estimation of derived length of such Lie solvable ideal is obtained depending on the class of Lie nilpotency of the Lie nilpotent one-sided ideal of $R.$ One-sided Lie nilpotent ideals contained in ideals generated by commutators of the form $[... [ [r_1, r_{2}], ... ], r_{n-1}], r_{n}]$ are also studied.
On graded u-nil clean rings
该文定义并研究分次 U-nil 洁净环，作为分次 nil-good 环的推广，并讨论其在矩阵环、分次群环及平凡环扩张中的性质转移。
Rings Whose Non-Units are Square-Nil Clean
本文研究强NUS-幂零清洁环（非单位元可表为交换的幂零元与平方幂等元之和），给出其判定准则，刻画仅含平凡幂等元时的等价条件，并提供矩阵构造与群环扩张的新例。
Nil-Essential Ideals
本文将本质理想推广为幂零本质理想，证明约化环中二者等价，并利用所有幂零本质理想的交等于基座这一事实，给出了环半单性的新判据。
Rings Whose Non-Invertible Elements are Weakly Nil-Clean
该研究在已有类幂零清洁环研究基础上，深入分析非单位元为弱幂零清洁的环的结构与刻画性质，并拓展研究了对应类型的群环。
Weakly Strongly 2-Nil-Clean Rings
本文引入弱强2-幂零-清洁环，证明其强π-正则且可分解为强2-幂零-清洁环与Z_{2^k}的直积。
Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I[x] for some ideals I of R.
On the Regularity and Clean Properties of Matrices over Dual Numbers
本文研究对偶数矩阵的正则性与洁净分解，通过相容性条件刻画冯·诺依曼正则性，并证明矩阵环 $M_n(\mathbb{D})$ 是强 $\pi$-正则且强洁净的，同时指出其非正则、非幂零洁净等性质。
Rings whose Non-Invertible Elements are Strongly Nil-Clean
本文研究非单位元分别为强诣零干净和唯一诣零干净的环，并证明这两类环可由已有环类完全刻画。
This work presents an example of a finitely presented semigroup $S$ containing an infinite nonnilpotent nil ideal $LS$, whose elements do not have a square (i.e. any word of the type $LXYYZ$ equals zero.)
We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L+rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also co…
The present paper introduces and studies some new types of rings and ideals such as completely nilary rings ( resp. completely nilary ideals ), weakly nilary ideals. Some properties of each are obtained and some characterizations of each type are given.
$m$-nil-clean nonderogatory matrices
本文证明在特征为奇素数 $p$ 的域上，当 $n\leq p$ 时，每个 $n\times n$ 非降阶矩阵若可表示为 $m$ 个幂等矩阵与一个幂零矩阵之和，则存在相应分解，且幂零部分满足特定幂零指数条件。
On Semi-Nil Clean Rings with Applications
本文提出并研究了半诣零清洁环、单位半诣零清洁环和 t-fine 环三类环，证明了半诣零清洁 NI 环必为周期环、群环半诣零清洁的充要条件以及 t-fine 环的矩阵环保持性质等结果。
Rings Whose Non-Invertible Elements Are Nil-Clean
本研究系统探讨非可逆元可表为幂等元加幂零元的环类，刻画其关键性质并推导结构结果，延续了该领域两篇经典文献的研究方向。
We show that the direct sum of an odd number of matrices $$C=\left(\begin{array}{cccc} 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&1 \end{array}\right)$$ cannot be a sum $P+Q$ of matrices over $\mathbb{F}_2$ satisfying $P^2=P$ and $Q^3=O$.
We introduce and investigate the so-called D-regularly nil clean rings by showing that these rings are, in fact, a non-trivial generalization of the classical von Neumann regular rings and of the strongly $π$-regular rings. Some other close relationships with certain well-known classes of rings such as $π$-regular rings, exchange rings, clean rings, nil-clean rings, etc., are also demonstrated. These results somewhat supply a recent publication of the author in Turk. J. Math. (2019) as well as they somewhat expand the important role of the two examples of nil-clean rings obtained by Ster in Li…
Rings Whose Non-Invertible Elements are Strongly Weakly Nil-Clean
本文研究非单位元素均为强弱幂零清洁的环，证明其为强π-正则，并给出基于Jacobson根和2×2矩阵环截断的完整刻画，同时探讨Morita上下文环和群环等扩展。
Motivated by the concept of clean ideals, we introduce the notion of weakly clean ideals. We define an ideal $I$ of a ring $R$ to be weakly clean ideal if for any $x\in I$, $x=u+e$ or $x=u-e$, where $u$ is a unit in $R$ and $e$ is an idempotent in $R$. We discuss various properties of weakly clean ideals.
A bound for the nilpotence index associated to $m$-nil-clean nonderogatory matrices
研究在特定条件下，$n\times n$非奇异矩阵$A$的分解问题，通过证明得出存在$m$个幂等矩阵和一个幂零矩阵使$A$可分解，且给出幂零矩阵幂零指数及$n > p$时$A$按$n$奇偶性的不同分解形式。
In a ring $A$ an ideal $I$ is called (principally) nilary if for any two (principal) ideals $V, W$ in $A$ with $VW\subseteq I,$ then either $V^n\subseteq I$ or $W^m\subseteq I,$ for some positive integers $m$ and $n$ depending on $V$ and $W;$ a ring $A$ is called (principally) nilary if the zero ideal is a (principally) nilary ideal~\cite{Birkenmeier2013133}. Let $G$ be a group and $A$ be a ring with unity. It is natural to ask when the group ring $A[G]$ is a (principally) nilary ring. We proved that, if $A[G]$ is a (principally) nilary ring, then the ring $A$ is a (principally) nilary ring; a…
Let R be a ring with the set of nilpotents Nil(R). We prove that the following are equivalent: (i) Nil(R) is additively closed, (ii) Nil(R) is multiplicatively closed and R satisfies Koethe's conjecture, (iii) Nil(R) is closed under the operation x\circ y=x+y-xy, (iv) Nil(R) is a subring of R. Some applications and examples of rings with this property are given, with an emphasis on certain classes of exchange and clean rings.
Rings Whose Invertible Elements Are Weakly Nil-Clean
本文研究所有可逆元均为弱幂清洁的环（UWNC环），证明三角矩阵环UWNC当且仅当底环UNC，并刻画了UWNC 2-素环及多项式环的UWNC性质。
Let R be a ring and let G be a group. We prove a rather curious necessary and sufficient condition for the commutative group ring RG to be weakly nil-neat only in terms of R,G and their sections. This somewhat expands three recent results, namely those established by McGovern et al. in (J. Algebra Appl., 2015), by Danchev-McGovern in (J. Algebra, 2015) and by the present authors in (J. Taibah Univ. Sci., 2020), related to commutative nil-clean, weakly nil-clean and nil-neat group rings, respectively.
We introduce a novel concept Frattinian nilpotent Lie algebra. Along with some examples, we show that every Frattinian nilpotent Lie algebra has a central decomposition of its ideals.
In this article, we introduce the weak ideal-Armendariz ring which combines Armendariz ring and weakly semicommutative properties of rings. In fact, it is a generalisation of an ideal-Armendariz ring. We investigate some properties of weak ideal Armendariz rings and prove that R is a weak ideal-Armendariz ring if and only if R[x] is weak ideal-Armendariz ring. Also, we generalise weak ideal-Armendariz as strongly nil-IFP and a number of properties are discussed which distinguishes it from other existing structures. We prove that if I is a semicommutative ideal of a ring R and R/I is a strongly…
We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is nil-clean for all $n\geq 1$ is paralleling to the corresponding implication for (Abelian, local) periodic rings. Besides, we study when the endomorphism ring $\mathrm{E}(G)$ of an Abelian group $G$ is periodic. Concretely, we establish that $\mathrm{E}(G)$ is periodic exactly when $G$ is finite as well as we find a complete necessary and sufficient condition when t…
Zero Insertive Nil Clean Rings
该文研究ZINC环的性质及与半交换、弱半交换环的关系，刻画了相关环的零插入元集合特征，证明了ZINC环直积、矩阵环的判定条件，并给出多项式扩张非ZINC的ZINC环实例。
Unit Uniquely Clean Rings
本文提出单位唯一清洁环（UnitUC），统一推广唯一清洁环与强诣零清洁环，并刻画其与布尔环、交换环及 Jacobson 根的关系，同时研究其在群环和矩阵环扩张下的表现。
On Some Extensions of $π$-Regular Rings
本文研究 $π$-正则与 nil clean 环变体的结构关系，证明 $(m,n)$-regularly nil clean 环左右对称及相关包含关系，并回答已有开放问题。
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