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 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.

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 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.

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 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.

USING IDEALS TO PROVIDE A UNIFIED APPROACH TO UNIQUELY CLEAN RINGS

2013 ◽  
Vol 96 (2) ◽  
pp. 258-274
Author(s):  
V. A. HIREMATH ◽  
SHARAD HEGDE
Keyword(s):  
Prime Ideal ◽  
Ideal Extension ◽  
Central Idempotent ◽  
Exchange Ring ◽  
Unified Approach ◽  
Clean Rings ◽  
Clean Ring ◽  
Nil Clean Ring

AbstractIn this article, we introduce the notion of the uniquely $I$-clean ring and show that, if $R$ is a ring and $I$ is an ideal of $R$ then $R$ is uniquely $I$-clean if and only if ($R/ I$ is Boolean and idempotents lift uniquely modulo $I$) if and only if (for each $a\in R$ there exists a central idempotent $e\in R$ such that $e- a\in I$ and $I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring $R$ and an ideal $I$ of $R$ under which uniquely $I$-clean rings coincide with uniquely clean rings. Further we prove that a ring $R$ is uniquely nil-clean if and only if ($N(R)$ is an ideal of $R$ and $R$ is uniquely $N(R)$-clean) if and only if $R$ is both uniquely clean and nil-clean if and only if ($R$ is an abelian exchange ring with $J(R)$ nil and every quasiregular element is uniquely clean). We also show that $R$ is a uniquely clean ring such that every prime ideal of $R$ is maximal if and only if $R$ is uniquely nil-clean ring and $N(R)= {\mathrm{Nil} }_{\ast } (R)$.

A note on commutative weakly nil clean rings

2016 ◽  
Vol 15 (10) ◽  
pp. 1620001 ◽  
Author(s):  
Alin Stancu
Keyword(s):  
Tensor Product ◽  
Sufficient Conditions ◽  
Clean Rings ◽  
Clean Ring ◽  
Nil Clean Ring

In this paper we discuss some properties of abelian (weakly) nil clean rings. We prove that any subring of an abelian (weakly) nil clean ring is (weakly) nil clean (Theorem 2). We also show that the tensor product of commutative (weakly) nil clean rings is also (weakly) nil clean and give sufficient conditions for the converse to be true (Theorems 3–6).

Nil-clean group rings

2016 ◽  
Vol 16 (07) ◽  
pp. 1750135 ◽  
Author(s):  
Serap Sahinkaya ◽  
Gaohua Tang ◽  
Yiqiang Zhou
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Group Ring ◽  
Nilpotent Element ◽  
General Situation ◽  
Group Rings ◽  
Locally Finite ◽  
Arbitrary Ring ◽  
Clean Ring ◽  
Nil Clean Ring

An element [Formula: see text] of a ring [Formula: see text] is nil-clean, if [Formula: see text], where [Formula: see text] and [Formula: see text] is a nilpotent element, and the ring [Formula: see text] is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl. 14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring [Formula: see text] and an abelian group [Formula: see text], the group ring [Formula: see text] is nil-clean, iff [Formula: see text] is nil-clean and [Formula: see text] is a [Formula: see text]-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite [Formula: see text]-group over a nil-clean ring is nil-clean, and that the hypercenter of the group [Formula: see text] must be a [Formula: see text]-group if a group ring of [Formula: see text] is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a [Formula: see text]-group.

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.

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.

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