Commutative feebly clean rings

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.2010 Mathematics Subject Classification. 16U99, 16E50, 13B99.

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“…This gives a contradiction. Thus proving (1). Let a ∈ R. In view of [4, Theorem 2.1], a ± a 2 ∈ N(R).…”

Section: Lemma 35 a Ring R Is Strongly Weakly Nil-clean If And Only Ifmentioning

confidence: 80%

“…A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. Commutative feebly clean rings were extensively investigated by [1] , motivated by the work on continuous function rings (see [1]). In this paper, strongly 2nil-clean rings are studied with an emphasis on their relations with feebly clean rings.…”

Section: ])mentioning

confidence: 99%

Rings additively generated by tripotents and nilpotents

Chen

Abdolyousefi

2019

J. Algebra Appl.

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“…For properties of weakly clean rings and connections with clean rings, we refer to [1,7,8,11]. Since several important properties of clean rings do not generalize to weakly clean rings, Arora and Kundu [3] introduced the family of feebly clean rings which has the property that every element r can be written as r = u+e 1 −e 2 , where u is a unit and e 1 , e 2 are orthogonal idempotents. Then both clean rings and weakly clean rings are feebly clean.…”

Section: Introductionmentioning

confidence: 99%

“…(b) ord n1 p = ϕ(n 1 )/2, ord n ′ m p = n ′ ord m p, and gcd(ord n1 p, ord n ′ m p) = 1. (c) ord n1 p = ϕ(n 1 ), ord n ′ m p = (n ′ ord m p)/2, and gcd(ord n1 p, ord n ′ m p) = 1 (note that in this case n ′ must be even) (3).…”

mentioning

confidence: 99%

On clean, weakly clean, and feebly clean commutative group rings

Li,

Zhong

2020

Preprint

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A ring R is said to be clean if each element of R can be written as the sum of a unit and an idempotent. R is said to be weakly clean if each element of R is either a sum or a difference of a unit and an idempotent, and R is said to be feebly clean if every element r can be written as r = u + e 1 − e 2 , where u is a unit and e 1 , e 2 are orthogonal idempotents. Clearly clean rings are weakly clean rings and both of them are feebly clean. In a recent article (J. Algebra Appl. 17 (2018), 1850111(5 pages)), McGoven characterized when the group ring Z (p) [Cq] is weakly clean and feebly clean, where p, q are distinct primes. In this paper, we consider a more general setting. Let K be an algebraic number field, O K its ring of integers, p ⊂ O a nonzero prime ideal, and Op the localization of O at p. We investigate when the group ring Op[G] is weakly clean and feebly clean, where G is a finite abelian group, and establish an explicit characterization for such a group ring to be weakly clean and feebly clean for the case when K = Q(ζn) is a cyclotomic field or K = Q( √ d) is a quadratic field.

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“…On the other vein, in order the enlarge in a non-trivial way the class of clean rings, in [1] were investigated the so-called feebly clean rings as rings R, each element r ∈ R being presentable like this: r = u + e − f , where u ∈ U (R) and e, f ∈ Id(R) with ef = f e = 0.…”

Section: Introductionmentioning

confidence: 99%