A note on uniquely (nil) clean ring

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...

Communications in Algebra, 2016

A *-ring R is called strongly nil *-clean if every element of R is the sum of a projection and a nilpotent element that commute with each other. In this paper we prove 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 with µ * = µ, η * = η ∈ R, the algebraic extension R[i] = {a + bi | a, b ∈ R, i 2 = µi + η } is strongly nil *-clean if and only if R is strongly nil *-clean and µη is nilpotent. We also prove that a *-ring R is commutative, strongly nil *-clean and every primary ideal is maximal if and only if every element of R is a projection.

TURKISH JOURNAL OF MATHEMATICS, 2021

The element q of a ring R is called quasi-idempotent element if q 2 = uq for some central unit u of R , or equivalently q = ue , where u is a central unit and e is an idempotent of R. In this paper, we define that the ring R is almost quasi-clean if each element of R is the sum of a regular element and a quasi-idempotent element. Several properties of almost-quasi clean rings are investigated. We prove that every quasi-continuous and nonsingular ring is almost quasi-clean. Finally, it is determined that the conditions under which the idealization of an R-module M is almost quasi clean.

Journal of Algebra, 2005

An associative ring with unity is called clean (respectively uniquely clean) if every element is (uniquely) the sum of an idempotent and a unit. In this paper we define clean general rings (with or without a unity) and extend many of the basic results to the wider class. In particular, a clean general ring is an exchange ring in the sense of Ara. We then study the general analogue of the uniquely clean rings and their relationship to the boolean rings. Finally, we introduce semiboolean rings as a natural generalization of boolean rings.  2005 Elsevier Inc. All rights reserved. and explored the notion of an exchange ring without unity. In this paper we extend the definition of a clean ring to general rings (without unity), and prove extensions of many of the results known for rings.

Miskolc Mathematical Notes, 2018

The conditions that allow an element of an associative, unital, not necessarily commutative ring R, to be represented as a sum of (commuting) idempotents and one nilpotent element are analyzed. Some applications to group rings are also presented.

Journal of Algebra and Its Applications

Question 3 of [3] asks whether the matrix ring [Formula: see text] is nil clean, for any nil clean ring [Formula: see text]. It is shown that, positive answer to this question is equivalent to positive solution for Köthe’s problem in the class of algebras over the field [Formula: see text]. 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.

Journal of Al-Qadisiyah for Computer Science and Mathematics

If all the elements of a ring W can be written as a sum of pure and idempotent elements", the ring is said to be P-clean. In this paper, we introduce the P-clean ring nation and look into some of its fundamental properties, examples, and relationship to the clean ring

A *-ring $R$ is called strongly nil *-clean if every element of $R$ is the sum of a projection and a nilpotent element that commute with each other. In this paper we investigate some properties of strongly nil *-rings and prove 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. We also prove that a *-ring $R$ is commutative, strongly nil *-clean and every primary ideal is maximal if and only if every element of R is a projection.

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 ...

Contemporary Mathematics, 2019

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.