From Boolean Rings to Clean Rings

A CLASS OF EXCHANGE RINGS

Glasgow Mathematical Journal ◽

10.1017/s0017089508004370 ◽

2008 ◽

Vol 50 (3) ◽

pp. 509-522 ◽

Cited By ~ 7

Author(s):

TSIU-KWEN LEE ◽

YIQIANG ZHOU

Keyword(s):

Exchange Ring ◽

Glasgow Math ◽

Basic Properties ◽

Boolean Rings ◽

Clean Rings ◽

Exchange Rings

AbstractIt is well known that a ring R is an exchange ring iff, for any a ∈ R, a−e ∈ (a2−a)R for some e2 = e ∈ R iff, for any a ∈ R, a−e ∈ R(a2−a) for some e2 = e ∈ R. The paper is devoted to a study of the rings R satisfying the condition that for each a ∈ R, a−e ∈ (a2−a)R for a unique e2 = e ∈ R. This condition is not left–right symmetric. The uniquely clean rings discussed in (W. K. Nicholson and Y. Zhou, Rings in which elements are uniquely the sum of an idempotent and a unit, Glasgow Math. J. 46 (2004), 227–236) satisfy this condition. These rings are characterized as the semi-boolean rings with a restricted commutativity for idempotents, where a ring R is semi-boolean iff R/J(R) is boolean and idempotents lift modulo J(R) (or equivalently, R is an exchange ring for which any non-zero idempotent is not the sum of two units). Various basic properties of these rings are developed, and a number of illustrative examples are given.

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Rings whose Elements are the Sum of a Tripotent and an Element from the Jacobson Radical

Canadian Mathematical Bulletin ◽

10.4153/s0008439519000092 ◽

2019 ◽

Vol 62 (4) ◽

pp. 810-821 ◽

Cited By ~ 1

Author(s):

M. Tamer Koşan ◽

Tülay Yildirim ◽

Y. Zhou

Keyword(s):

Abelian Group ◽

Nilpotent Group ◽

Group Ring ◽

Jacobson Radical ◽

Sufficient Conditions ◽

Morita Context ◽

Locally Finite ◽

Boolean Rings ◽

Clean Rings ◽

Necessary And Sufficient

AbstractThis paper is about rings $R$ for which every element is a sum of a tripotent and an element from the Jacobson radical $J(R)$. These rings are called semi-tripotent rings. Examples include Boolean rings, strongly nil-clean rings, strongly 2-nil-clean rings, and semi-boolean rings. Here, many characterizations of semi-tripotent rings are obtained. Necessary and sufficient conditions for a Morita context (respectively, for a group ring of an abelian group or a locally finite nilpotent group) to be semi-tripotent are proved.

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Ideally nil clean rings

Communications in Algebra ◽

10.1080/00927872.2021.1929276 ◽

2021 ◽

pp. 1-12

Author(s):

Alexi Block Gorman ◽

Alexander Diesl

Keyword(s):

Clean Rings

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Nil-clean quadratic elements

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501973 ◽

2017 ◽

Vol 16 (10) ◽

pp. 1750197 ◽

Cited By ~ 2

Author(s):

Janez Šter

Keyword(s):

Complete Classification ◽

Strong Condition ◽

Clean Rings

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

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Strongly nil regular clean rings

International Mathematical Forum ◽

10.12988/imf.2021.912248 ◽

2021 ◽

Vol 16 (3) ◽

pp. 125-129

Author(s):

Zubaida M. Ibraheem ◽

Raghad I. Zidan

Keyword(s):

Clean Rings

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Rings additively generated by tripotents and nilpotents

Journal of Algebra and Its Applications ◽

10.1142/s0219498819500506 ◽

2019 ◽

Vol 18 (03) ◽

pp. 1950050

Author(s):

Huanyin Chen ◽

Marjan Sheibani Abdolyousefi

Keyword(s):

Clean Rings ◽

Unit Formula

A ring [Formula: see text] is Zhou nil-clean if every element in [Formula: see text] is the sum of two tripotents and a nilpotent that commute. A ring [Formula: see text] is feebly clean if for any [Formula: see text] there exist two orthogonal idempotents [Formula: see text] and a unit [Formula: see text] such that [Formula: see text]. In this paper, Zhou nil-clean rings are further discussed with an emphasis on their relations with feebly clean rings. We prove that a ring [Formula: see text] is Zhou nil-clean if and only if [Formula: see text] is feebly clean, [Formula: see text] is nil and [Formula: see text] has exponent [Formula: see text] if and only if [Formula: see text] is weakly exchange, [Formula: see text] is nil and [Formula: see text] has exponent [Formula: see text]. New properties of Zhou rings are thereby obtained.

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