Strongly nil * -clean rings

arXiv (Cornell University), 2024

An element x ∈ R is considered (strongly) nil-clean if it can be expressed as the sum of an idempotent e ∈ R and a nilpotent b ∈ R (where eb = be). If for any x ∈ R, there exists a unit u ∈ R such that ux is (strongly) nil-clean, then R is called a (strongly) unit nil-clean ring. It is worth noting that any unit-regular ring is strongly unit nil-clean. In this note, we provide a characterization of the unit regularity of a group ring, along with an additional condition. We also fully characterize the unitregularity of the group ring Z n G for every n > 1. Additionally, we discuss strongly unit nil-cleanness in the context of Morita contexts, matrix rings, and group rings.

2015

summary:In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let $J^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called \emph {very $J^{\#}$-clean} provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of $J^{\#}$. A ring $R$ is said to be \emph {very $J^{\#}$-clean} in case every element in $R$ is very $J^{\#}$-clean. We prove that every very $J^{\#}$-clean ring is strongly $\pi $-rad clean and has stable range one. It is shown that for a commutative local ring $R$, $A(x)\in M_2\big (R[[x]]\big )$ is very $J^{\#}$-clean if and only if $A(0)\in M_2(R)$ is very $J^{\#}$-clean. Various basic characterizations and properties of these rings are proved. We obtain a partial answer to the open question whether strongly clean rings have stable range one. This paper is dedicated to Professor Abdullah Harmanci on...

Linear Algebra and its Applications, 2007

A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute with each other. For a commutative local ring R and for an arbitrary integer n 2, the paper deals with the question whether the strongly clean property of

arXiv (Cornell University), 2024

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 work somewhat continues two publications on the subject due to Diesl (J.

Journal of Algebra and Its Applications, 2015

Generalizing the notion of nil cleanness from , in parallel to , we define the concept of 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 weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

International Mathematical Forum, 2011

Throughout this paper, all rings are commutative with unity. For a ring R, Id(R), U(R) and J(R) are denote the set of idempotents of R , the set of units of R and the Jacobson radical of R, respectively. Nicholson [11] defined a ring R to be clean if every element of R can be ...

Journal of Algebra and Its Applications, 2012

In [A. J. Diesl, Classes of strongly clean rings, Ph.D. Dissertation, University of California, Berkely (2006); Nil clean rings, J. Algebra383 (2013) 197–211], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short paper, we characterize nil clean commutative group rings.

Erzincan University Journal of Science and Technology, 2022

In this paper our goal to thoroughly determine the rings in which each non-unit element is a product of a nilpotent and a quasi-idempotent.

2013

A ring with an involution * is called strongly J-*-clean if every element is a sum of a projection and an element of the Jacobson radical that commute. In this article, we prove several results characterizing this class of rings. It is shown that a *-ring R is strongly J-*-clean, if and only if R is uniquely clean and strongly *-clean, if and only if R is uniquely strongly *-clean, that is, for any a∈ R, there exists a unique projection e∈ R such that a-e is invertible and ae=ea.

Cornell University - arXiv, 2017

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. We prove that a ring R is a Zhou nil-clean ring if and only if for any a ∈ R, there exists e ∈ Z[a] such that a − e ∈ R is nilpotent and e 5 = 5e 3 − 4e, if and only if for any a ∈ R, there exist idempotents e, f, g, h ∈ Z[a] and a nilpotent w such that a = e + f + g + h + w, if and only if for any a ∈ R, there exist 2-idempotents e, f ∈ Z[a] and a nilpotent w ∈ R such that a = e + f + w, if and only if for any a ∈ R, there exists a 2-idempotent e ∈ Z[a] and a nilpotent w ∈ R such that a 2 = e + w, if and only if R is a Kosan exchange ring. As corollaries, new characterizations of strongly 2-nil-clean rings are thereby obtained.