scholarly journals On  mj mj -Clean Ring and Strongly  mj mj -Clean Ring

2021 ◽  
Keyword(s):  
Clean Ring

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.

scholarly journals Rings in which every element is either a sum or a difference of a nilpotent and an idempotent

2016 ◽  
Vol 15 (08) ◽  
pp. 1650148 ◽  
Author(s):  
Simion Breaz ◽  
Peter Danchev ◽  
Yiqiang Zhou
Keyword(s):  
Division Ring ◽  
Matrix Ring ◽  
Decomposition Theorem ◽  
Arbitrary Ring ◽  
Clean Rings ◽  
Clean Ring ◽  
Unital Rings ◽  
Nil Clean Ring ◽  
Comprehensive Study

Generalizing the notion of nil-cleanness from [A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211], in parallel to [P. V. Danchev and W. Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra 425 (2015) 410–422], we define the concept of weak nil-cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

CLEAN, ALMOST CLEAN, POTENT COMMUTATIVE RINGS

2007 ◽  
Vol 06 (04) ◽  
pp. 671-685 ◽  
Author(s):  
K. VARADARAJAN
Keyword(s):  
Commutative Ring ◽  
Analogous Result ◽  
Commutative Rings ◽  
Complete Characterization ◽  
Zariski Topology ◽  
Clean Ring ◽  
Regular Rings

We give a complete characterization of the class of commutative rings R possessing the property that Spec(R) is weakly 0-dimensional. They turn out to be the same as strongly π-regular rings. We considerably strengthen the results of K. Samei [13] tying up cleanness of R with the zero dimensionality of Max(R) in the Zariski topology. In the class of rings C(X), W. Wm Mc Govern [6] has characterized potent rings as the ones with X admitting a clopen π-base. We prove the analogous result for any commutative ring in terms of the Zariski topology on Max(R). Mc Govern also introduced the concept of an almost clean ring and proved that C(X) is almost clean if and only if it is clean. We prove a similar result for all Gelfand rings R with J(R) = 0.

Rings in which elements are sums of tripotents and nilpotents

2018 ◽  
Vol 17 (03) ◽  
pp. 1850042 ◽  
Author(s):  
Marjan Sheibani Abdolyousefi ◽  
Huanyin Chen
Keyword(s):  
Clean Ring ◽  
Nil Clean Ring

A ring [Formula: see text] is strongly 2-nil-clean if every element in [Formula: see text] is the sum of a tripotent and a nilpotent that commute. We prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if [Formula: see text] is a strongly feebly clean 2-UU ring if and only if [Formula: see text] is an exchange 2-UU ring. Furthermore, we characterize strongly 2-nil-clean ring via involutions. We show that a ring [Formula: see text] is strongly 2-nil-clean if and only if every element in [Formula: see text] is the sum of an idempotent, an involution and a nilpotent that commute.

On *-clean group rings

2014 ◽  
Vol 14 (01) ◽  
pp. 1550004 ◽  
Author(s):  
Yuanlin Li ◽  
M. M. Parmenter ◽  
Pingzhi Yuan
Keyword(s):  
Local Ring ◽  
Prime Order ◽  
Sufficient Conditions ◽  
Complete Characterization ◽  
Group Rings ◽  
Clean Ring ◽  
Commutative Local Ring ◽  
Irreducible Factorization ◽  
Necessary And Sufficient

A ring with involution * is called *-clean if each of its elements is the sum of a unit and a projection. Clearly a *-clean ring is clean. Vaš asked whether there exists a clean ring with involution * that is not *-clean. In a recent paper, Gao, Chen and the first author investigated when a group ring RG with classical involution * is *-clean and obtained necessary and sufficient conditions for RG to be *-clean, where R is a commutative local ring and G is one of C3, C4, S3 and Q8. As a consequence, the authors provided many examples of group rings which are clean, but not *-clean. In this paper, we continue this investigation and we give a complete characterization of when the group algebra 𝔽Cp is *-clean, where 𝔽 is a field and Cp is the cyclic group of prime order p. Our main result is related closely to the irreducible factorization of a pth cyclotomic polynomial over the field 𝔽. Among other results we also obtain a complete characterization of when RCn (3 ≤ n ≤ 6) is *-clean where R is a commutative local ring.

Corner Rings of a Clean Ring Need Not be Clean

2012 ◽  
Vol 40 (5) ◽  
pp. 1595-1604 ◽  
Author(s):  
Janez Šter
Keyword(s):  
Clean Ring

scholarly journals Abel Rings and Super-Strongly Clean Rings

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinchun Qu ◽  
Junchao Wei
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Clean Rings ◽  
Clean Ring ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

Abstract In this note, we first show that a ring R is Abel if and only if the 2 × 2 upper triangular matrix ring over R is quasi-normal. Next, we give the notion of super-strongly clean ring (that is, an Abel clean ring), which is inbetween uniquely clean rings and strongly clean rings. Some characterizations of super-strongly clean rings are given.

Nil-Clean Rings with Involution

2021 ◽  
Vol 28 (03) ◽  
pp. 367-378
Author(s):  
Jian Cui ◽  
Guoli Xia ◽  
Yiqiang Zhou
Keyword(s):  
Boolean Ring ◽  
Matrix Rings ◽  
Clean Rings ◽  
Clean Ring ◽  
Rings With Involution ◽  
Text Version

A [Formula: see text]-ring [Formula: see text] is called a nil [Formula: see text]-clean ring if every element of [Formula: see text] is a sum of a projection and a nilpotent. Nil [Formula: see text]-clean rings are the [Formula: see text]-version of nil-clean rings introduced by Diesl. This paper is about the nil [Formula: see text]-clean property of rings with emphasis on matrix rings. We show that a [Formula: see text]-ring [Formula: see text] is nil [Formula: see text]-clean if and only if [Formula: see text] is nil and [Formula: see text] is nil [Formula: see text]-clean. For a 2-primal [Formula: see text]-ring [Formula: see text], with the induced involution given by[Formula: see text], the nil [Formula: see text]-clean property of [Formula: see text] is completely reduced to that of [Formula: see text]. Consequently, [Formula: see text] is not a nil [Formula: see text]-clean ring for [Formula: see text], and [Formula: see text] is a nil [Formula: see text]-clean ring if and only if [Formula: see text] is nil, [Formula: see text]is a Boolean ring and [Formula: see text] for all [Formula: see text].

A nil-clean 2 × 2 matrix over the integers which is not clean

2014 ◽  
Vol 13 (06) ◽  
pp. 1450009 ◽  
Author(s):  
Dorin Andrica ◽  
Grigore Călugăreanu
Keyword(s):  
Matrix Ring ◽  
Clean Ring ◽  
The Matrix ◽  
Nil Clean Ring ◽  
Clean Element

While any nil-clean ring is clean, the last eight years, it was not known whether nil-clean elements in a ring are clean. We give an example of nil-clean element in the matrix ring ℳ2(Z) which is not clean.

scholarly journals ON CLEAN LAURENT SERIES RINGS

2013 ◽  
Vol 95 (3) ◽  
pp. 421-427 ◽  
Author(s):  
YIQIANG ZHOU ◽  
MICHAŁ ZIEMBOWSKI
Keyword(s):  
Laurent Series ◽  
Series Ring ◽  
Clean Ring

AbstractHere we prove that, for a $2$-primal ring $R$, the Laurent series ring $R((x))$ is a clean ring if and only if $R$ is a semiregular ring with $J(R)$ nil. This disproves the claim in K. I. Sonin [‘Semiprime and semiperfect rings of Laurent series’, Math. Notes 60 (1996), 222–226] that the Laurent series ring over a clean ring is again clean. As an application of the result, it is shown that, for a $2$-primal ring $R$, $R((x))$ is semiperfect if and only if $R((x))$ is semiregular if and only if $R$ is semiperfect with $J(R)$ nil.

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