A ring $R$ is called clean if every element of $R$ is the sum of a unit and an idempotent. Motivated by a question proposed by Lam on the cleanness of von Neumann Algebras, Vaš introduced a more natural concept of cleanness for $*$-rings, called the $*$-cleanness. More precisely, a $*$-ring $R$ is called a $*$-clean ring if every element of $R$ is the sum of a unit and a projection ($*$-invariant idempotent). Let $\mathbb F$ be a finite field and $G$ a finite abelian group. In this paper, we introduce two classes of involutions on group rings of the form $\mathbb FG$ and characterize the $*$-cleanness of these group rings in each case. When $*$ is taken as the classical involution, we also characterize the $*$-cleanness of $\mathbb F_qG$ in terms of LCD abelian codes and self-orthogonal abelian codes in $\mathbb F_qG$.
Two questions of L. Va${\rm \check{\textbf{s}}}$ on *-clean rings
一个具有对合 * 的环$R$称为（强）*-clean，如果$R$的每个元素都是单位元与投影（可交换的）之和.所有的戒指都是干净的。Va ${\rm \check{s}}$ [L. Va ${\rm \check{s}}$，*-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras，J. Algebra 324（12）（2010）3388-3400]询问是否存在干净但不干净的 *-环以及单位正则和 *-正则环是否强 *-clean。在本文中，我们回答这两个问题的几个例子。此外，还给出了单位正则环和 *-正则环的一些刻划.
*-Clean Rings; Some Clean and Almost Clean Baer *-rings and von Neumann Algebras
戒指是干净的（resp。几乎干净），如果它的每个元素都是一个单位的总和（分别是）。正则元素）和幂等元。在本文中，我们定义了类似的概念，*-环：一个 *-环是 *-干净的（分别）。almost *-clean），如果它的每个元素都是一个单位的和（分别是规则元素）和投影。虽然 *-clean是一个比clean更强的概念，但对于某些 *-环，我们证明它更自然。单位正则环的清洁性定理[V. P. Camillo，D. Khurana，A Characterization of Unit正则环，Communications in Algebra，29（5）（2001）2293-2295]被修改为阿贝尔（或约化或Armendariz）的 *-正则环的 *-清洁性。利用这个结果，证明了所有满足某些公理的有限的I型Baer *-环（在[S. K. Berberian，Baer *-rings，Die Grundlehren der mathematischen Wissenschaften 195，Springer-Verlag，Berlin-Heidelberg-New York，1972]和[L. Vas，Dimension and Torsion Theories for a Class of Baer *-Rings，Journal of Algebra，289（2）（2005）614-6…
Let n be an arbitrary natural number. The class of (strongly) n-torsion clean rings is introduced and investigated. Abelian n-torsion clean rings are somewhat characterized and a complete characterization of strongly n-torsion clean rings is given in the case when n is odd. Some open questions are posed at the end.
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.
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 lifted uniquely weakly modulo $I$ if and only if for each $a\in R$, there exists a central idempotent $e\in R$ such that either $a-e\in I$ or $a+e\in I$ and $I$ is idempotent free. As a corollary, we characterize weakly $J$-clean ring. Also we study various properties of weakly $I$-clean ring.
受环的clean指标概念的启发，引入了环的弱clean指标概念。对于具有单位元的环$R$的任何元素$a$，我们定义了$ χ(a)=\{e\in R\mid e^2=e\text{ and }a-e \mbox{ or } a+e \mbox{ is a unit}\}$。$R$的弱清洁指数被定义为$\sup \{|χ(a)|: a\in R\}$，并且它被表示为$\win(R),$，其中$| χ(a)| $表示集合$χ(a)$的基数。本文刻画了弱clean指标环$1$，$2$和$3$ .
We completely determine those natural numbers $n$ for which the full matrix ring $M_n(F_2)$ and the triangular matrix ring $T_n(F_2)$ over the two elements field $F_2$ are either n-torsion clean or are almost n-torsion clean, respectively. These results somewhat address and settle a question, recently posed by Danchev-Matczuk in Contemp. Math. (2019) as well as they supply in a more precise aspect the nil-cleanness property of the full matrix $n\times n$ ring $M_n(F_2)$ for all naturals $n \geq 1$, established in Linear Algebra and Appl. (2013) by Breaz-Calugareanu-Danchev-Micu and again in Li…
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 when a $2\times 2$ matrix over a commutative local ring is strongly rad-clean.
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
Rings Whose Non-Invertible Elements Are Uniquely Strongly Clean
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Shahed University（沙赫德大学）；Tarbiat Modares University（塔比尔·莫达雷斯大学）
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 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…
