n-Clean Rings

2006 ◽  
Vol 13 (04) ◽  
pp. 599-606 ◽  
Author(s):  
Guangshi Xiao ◽  
Wenting Tong
Keyword(s):  
Positive Integer ◽  
Polynomial Ring ◽  
Matrix Ring ◽  
Clean Rings ◽  
Clean Ring

Let n be a positive integer. A ring R is called n-clean if every element of R can be written as a sum of an idempotent and n units in R. The class of n-clean rings contains clean rings and (S,n)-rings (i.e., every element is a sum of no more than n units). In this paper, we investigate some properties on n-clean rings. There exists a clean and (S,3)-ring which is not an (S,2)-ring. If R is a ring satisfying (SI), then the polynomial ring R[x] is not n-clean for any positive integer n. An example shows that for any positive integer n> 1, there exists a non n-clean ring R such that the 2× 2 matrix ring M2(R) over R is n-clean.

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.

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.

scholarly journals Conjugate (nil) clean rings and Köthe’s problem

2017 ◽  
Vol 16 (04) ◽  
pp. 1750073 ◽  
Author(s):  
Jerzy Matczuk
Keyword(s):  
Positive Solution ◽  
Matrix Ring ◽  
Positive Answer ◽  
Clean Rings ◽  
Clean Ring ◽  
The Matrix ◽  
Equivalent Problems ◽  
Nil Clean Ring

Question 3 of [3] asks whether the matrix ring [Formula: see text] is nil clean, for any nil clean ring [Formula: see text]. It is shown that, positive answer to this question is equivalent to positive solution for Köthe’s problem in the class of algebras over the field [Formula: see text]. Other equivalent problems are also discussed. The classes of conjugate clean and conjugate nil clean rings, which lie strictly between uniquely (nil) clean and (nil) clean rings are introduced and investigated.

scholarly journals On two open problems about strongly clean rings

2004 ◽  
Vol 70 (2) ◽  
pp. 279-282 ◽  
Author(s):  
Zhou Wang ◽  
Jianlong Chen
Keyword(s):  
Matrix Ring ◽  
Triangular Matrix ◽  
Open Problems ◽  
Clean Rings ◽  
Clean Ring ◽  
The Matrix ◽  
Triangular Matrix Ring ◽  
Semiperfect Ring ◽  
Upper Triangular Matrix ◽  
Upper Triangular Matrix Ring

A ring is called strongly clean if every element is the sum of an idempotent and a unit which commute. In 1999 Nicholson asked whether every semiperfect ring is strongly clean and whether the matrix ring of a strongly clean ring is strongly clean. In this paper, we prove that if R = {m/n ∈ ℚ: n is odd}, then M2(R) is a semiperfect ring but not strongly clean. Thus, we give negative answers to both questions. It is also proved that every upper triangular matrix ring over the ring R is strongly clean.

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.

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.

scholarly journals On Segre products of affine semigroup rings

1988 ◽  
Vol 110 ◽  
pp. 113-128 ◽  
Author(s):  
Lê Tuân Hoa
Keyword(s):  
Positive Integer ◽  
Polynomial Ring ◽  
Semigroup Ring ◽  
Semigroup Rings ◽  
Finitely Generated ◽  
Affine Semigroup

Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submonoid S of the additive monoid Nm for some positive integer m. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t1, …, tm] generated by the monomials .

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 γ-Clean Rings

2017 ◽  
Vol 5 (3) ◽  
pp. 285
Author(s):  
Shaimaa S. Esa ◽  
Hewa S. Faris
Keyword(s):  
Basic Properties ◽  
Clean Rings ◽  
Clean Ring

In this paper we introduce the concept of -clean ring and we discuss some relations between - clean ring and other rings with explaining by some examples. Also, we give some basic properties of it.

Sign in / Sign up

Export Citation Format

Share Document