A ring $R$ is a strongly clean ring if every element in $R$ is the sum of an idempotent and a unit that commutate. We construct some classes of strongly clean rings which have stable range one. It is shown that such cleanness of $2 imes 2$ matrices over commutative local rings is completely determined in terms of solvability of quadratic equations.

An element $a$ in a ring $R$ is very clean in case there exists‎ ‎an idempotent $ein R$ such that $ae = ea$ and either $a‎- ‎e$ or $a‎ + ‎e$ is invertible‎. ‎An element $a$ in a ring $R$ is very $J$-clean‎ ‎provided that there exists an idempotent $ein R$ such that $ae =‎ ‎ea$ and either $a-ein J(R)$ or $a‎ + ‎ein J(R)$‎. ‎Let $R$ be a‎ ‎local ring‎, ‎and let $sin C(R)$‎. ‎We prove that $Ain K_...

In this paper, we introduce the new notion of n-f-clean rings as a generalization of f-clean rings. Next, we investigate some properties of such rings. We prove that $M_n(R)$ is n-f-clean for any n-f-clean ring R. We also, get a condition under which the denitions of n-cleanness and n-f-cleanness are equivalent.

in this paper, we introduce the new notion of strongly j-clean rings associatedwith polynomial identity g(x) = 0, as a generalization of strongly j-clean rings. we denotestrongly j-clean rings associated with polynomial identity g(x) = 0 by strongly g(x)-j-cleanrings. next, we investigate some properties of strongly g(x)-j-clean.

A ring R is said to be n-clean if every element can be written as a sum of an idempotent and n units. The class of these rings contains clean ring and n-good rings in which each element is a sum of n units. In this paper, we show that for any ring R, the endomorphism ring of a free R-module of rank at least 2 is 2-clean and that the ring B(R) of all ω × ω row and column-finite matrices over any...

Let R be an Abelian exchange ring. We prove the following results: 1. RZ2 and RS3 are clean rings. 2. If G is a group of prime order p and p is in the Jacobson radical of R, then RG is clean. 3. If identity in R is a sum of two units and G is a locally finite group, then every element in RG is a sum of two units. 4. For any locally finite group G, RG has stable range one. All rings in this note...

We introduce the notion of clean ideal, which is a natural generalization of clean rings. It is shown that every matrix ideal over a clean ideal of a ring is clean. Also we prove that every ideal having stable range one of a regular ring is clean. These generalize the corresponding results for clean rings. 1. Introduction. Let R be a unital ring. We say that R is a clean ring in case every elem...

let a; b; k 2 k and u ; v 2 u(k). we show for any idempotent e 2 k, ( a 0 b 0 ) is e-clean i ( a 0 u(vb + ka) 0 ) is e-clean and if ( a 0 b 0 ) is 0-clean, ( ua 0 u(vb + ka) 0 ) is too.

We show that if $R$ is a ring with an arbitrary idempotent $e$ such that $eRe$ and $(1-e)R(1-e)$ are both strongly nil-clean rings‎, ‎then $R/J(R)$ is nil-clean‎. ‎In particular‎, ‎under certain additional circumstances‎, ‎$R$ is also nil-clean‎. ‎These results somewhat improves on achievements due to Diesl in J‎. ‎Algebra (2013) and to Koc{s}an-Wang-Zhou in J‎. ‎Pure Appl‎. ‎Algebra (2016)‎. ‎...

Let R be a commutative local ring. It is proved that R is Henselian if and only if each R-algebra which is a direct limit of module finite R-algebras is strongly clean. So, the matrix ring Mn(R) is strongly clean for each integer n > 0 if R is Henselian and we show that the converse holds if either the residue class field of R is algebraically closed or R is an integrally closed domain or R is ...