Rings additively generated by tripotents and nilpotents

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.

Quasi-clean rings and strongly quasi-clean rings

2021 ◽  
pp. 2150079
Author(s):  
Gaohua Tang ◽  
Huadong Su ◽  
Pingzhi Yuan
Keyword(s):  
Positive Integer ◽  
Natural Generalization ◽  
Extensive Study ◽  
Central Unit ◽  
Boolean Ring ◽  
Semilocal Ring ◽  
Clean Rings ◽  
Maximal Ideals ◽  
Open Questions ◽  
Unit Formula

An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see text] is called a quasi-Boolean ring if every element of [Formula: see text] is quasi-idempotent. A ring [Formula: see text] is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or [Formula: see text] has no image isomorphic to [Formula: see text]; For an indecomposable commutative semilocal ring [Formula: see text] with at least two maximal ideals, [Formula: see text]([Formula: see text]) is strongly quasi-clean if and only if [Formula: see text] is quasi-clean if and only if [Formula: see text], [Formula: see text] is a maximal ideal of [Formula: see text]. For a prime [Formula: see text] and a positive integer [Formula: see text], [Formula: see text] is strongly quasi-clean if and only if [Formula: see text]. Some open questions are also posed.

Strongly unit nil-clean rings

2017 ◽  
Vol 16 (06) ◽  
pp. 1750115
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani
Keyword(s):  
Nilpotent Element ◽  
Matrix Rings ◽  
Clean Rings ◽  
Clean Element ◽  
Unit Formula

An element in a ring is strongly nil-clean, if it is the sum of an idempotent and a nilpotent element that commute. A ring [Formula: see text] is strongly unit nil-clean, if for any [Formula: see text] there exists a unit [Formula: see text], such that [Formula: see text] is strongly nil-clean. We prove, in this paper, that a ring [Formula: see text] is strongly unit nil-clean, if and only if every element in [Formula: see text] is equivalent to a strongly nil-clean element, if and only if for any [Formula: see text], there exists a unit [Formula: see text], such that [Formula: see text] is strongly [Formula: see text]-regular. Strongly unit nil-clean matrix rings are investigated as well.

Ideally nil clean rings

2021 ◽  
pp. 1-12
Author(s):  
Alexi Block Gorman ◽  
Alexander Diesl
Keyword(s):  
Clean Rings

scholarly journals Nil-clean quadratic elements

2017 ◽  
Vol 16 (10) ◽  
pp. 1750197 ◽  
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].

Strongly nil regular clean rings

2021 ◽  
Vol 16 (3) ◽  
pp. 125-129
Author(s):  
Zubaida M. Ibraheem ◽  
Raghad I. Zidan
Keyword(s):  
Clean Rings

On some punctured codes of ℤq-Simplex codes

2021 ◽  
pp. 2250012
Author(s):  
J. Prabu ◽  
J. Mahalakshmi ◽  
C. Durairajan ◽  
S. Santhakumar
Keyword(s):  
Prime Divisor ◽  
Weight Distribution ◽  
Linear Code ◽  
Prime Power ◽  
Punctured Codes ◽  
New Codes ◽  
Simplex Codes ◽  
Simplex Code ◽  
Partial Weight ◽  
Unit Formula

In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

scholarly journals On f-clean rings and f-clean elements

2011 ◽  
Vol 30 (2) ◽  
pp. 277-284 ◽  
Author(s):  
Ali H Handam
Keyword(s):  
Clean Rings

The crystal structure of gatehouseite

2011 ◽  
Vol 75 (6) ◽  
pp. 2823-2832
Author(s):  
P. Elliott ◽  
A. Pring
Keyword(s):  
Crystal Structure ◽  
Hydrogen Bonding ◽  
Three Dimensional ◽  
Direct Methods ◽  
Chemical Analyses ◽  
X Ray Diffraction ◽  
X Ray ◽  
Manganese Phosphate ◽  
Phosphate Mineral ◽  
Unit Formula

AbstractThe crystal structure of the manganese phosphate mineral gatehouseite, ideally Mn52+(PO4)2(OH)4, space group P212121, a = 17.9733(18), b = 5.6916(11), c = 9.130(4) Å, V= 933.9(4) Å3, Z = 4, has been solved by direct methods and refined from single-crystal X-ray diffraction data (T = 293 K) to an R index of 3.76%. Gatehouseite is isostructural with arsenoclasite and with synthetic Mn52+(PO4)2(OH)4. The structure contains five octahedrally coordinated Mn sites, occupied by Mn plus very minor Mg with observed <Mn—O> distances from 2.163 to 2.239 Å. Two tetrahedrally coordinated P sites, occupied by P, Si and As, have <P—O> distances of 1.559 and 1.558 Å. The structure comprises two types of building unit. A strip of edge-sharing Mn(O,OH)6 octahedra, alternately one and two octahedra wide, extends along [010]. Chains of edge- and corner-shared Mn(O,OH)6 octahedra coupled by PO4 tetrahedra extend along [010]. By sharing octahedron and tetrahedron corners, these two units form a dense three-dimensional framework, which is further strengthened by weak hydrogen bonding. Chemical analyses by electron microprobe gave a unit formula of (Mn4.99Mg0.02)Σ5.01(P1.76Si0.07(As0.07)Σ2.03O8(OH)3.97.

scholarly journals WEAKLY NIL-CLEAN INDEX AND UNIQUELY WEAKLY NIL-CLEAN RINGS

2017 ◽  
Vol 21 (21) ◽  
pp. 180-180 ◽  
Author(s):  
Andrada Ciımpean ◽  
Peter Danchev
Keyword(s):  
Clean Rings ◽  
Clean Index

Commutative nil clean group rings

2015 ◽  
Vol 14 (06) ◽  
pp. 1550094 ◽  
Author(s):  
Warren Wm. McGovern ◽  
Shan Raja ◽  
Alden Sharp
Keyword(s):  
Commutative Group ◽  
University Of California ◽  
Short Paper ◽  
Group Rings ◽  
Clean Rings ◽  
Clean Ring ◽  
Nil Clean Ring

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.

Sign in / Sign up

Export Citation Format

Share Document