Structure of Zhou Nil-clean Rings

2018 ◽  
Vol 25 (03) ◽  
pp. 361-368
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani
Keyword(s):  
Direct Product ◽  
Matrix Ring ◽  
Boolean Ring ◽  
Clean Rings ◽  
Generalized Matrix ◽  
Bell Ring

A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ϵ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM(R) is 5-potent and BM(R) is nil, if and only if J(R) is nil and R/J(R) is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn(R; s) is Zhou nil-clean if and only if R is Zhou nil-clean and s ϵ J(R).

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.

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 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].

scholarly journals Classical quotient rings of generalized matrix rings

1995 ◽  
Vol 18 (2) ◽  
pp. 311-316 ◽  
Author(s):  
David G. Poole ◽  
Patrick N. Stewart
Keyword(s):  
Associative Ring ◽  
Matrix Ring ◽  
Quotient Ring ◽  
Matrix Rings ◽  
Finite Set ◽  
Generalized Matrix ◽  
Quotient Rings

An associative ringRwith identity is a generalized matrix ring with idempotent setEifEis a finite set of orthogonal idempotents ofRwhose sum is1. We show that, in the presence of certain annihilator conditions, such a ring is semiprime right Goldie if and only ifeReis semiprime right Goldie for alle∈E, and we calculate the classical right quotient ring ofR.

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.

Rings whose cyclics are C3-modules

2016 ◽  
Vol 15 (08) ◽  
pp. 1650152 ◽  
Author(s):  
Yasser Ibrahim ◽  
Xuan Hau Nguyen ◽  
Mohamed F. Yousif ◽  
Yiqiang Zhou
Keyword(s):  
Direct Product ◽  
Regular Ring ◽  
Matrix Ring ◽  
Artinian Ring ◽  
Local Rings ◽  
Classical Theorem ◽  
Basic Properties ◽  
Semiperfect Ring ◽  
Injectivity Condition ◽  
Strongly Regular

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or [Formula: see text]. Here we carry out a study of the rings whose cyclic modules are [Formula: see text]-modules. The motivation is the observation that a ring [Formula: see text] is semisimple artinian if and only if every [Formula: see text] -generated right [Formula: see text]-module is a [Formula: see text]-module. Many basic properties are obtained for the rings whose cyclics are [Formula: see text]-modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics [Formula: see text]-modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a [Formula: see text] matrix ring over a strongly regular ring. Applications to the rings whose [Formula: see text]-generated modules are [Formula: see text] -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed.

The group K 0 of a generalized matrix ring

2013 ◽  
Vol 52 (3) ◽  
pp. 250-261 ◽  
Author(s):  
P. A. Krylov
Keyword(s):  
Matrix Ring ◽  
Generalized Matrix

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.

Quasi-Baer and biregular generalized matrix rings

2017 ◽  
Vol 16 (04) ◽  
pp. 1750067 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Donald D. Davis
Keyword(s):  
Principal Ideal ◽  
Matrix Ring ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Principal Ideals ◽  
Generalized Matrix ◽  
The Right

Generalized matrix rings are ubiquitous in algebra and have relevant applications to analysis. A ring is quasi-Baer (respectively, right p.q.-Baer) in case the right annihilator of any ideal (respectively, principal ideal) is generated by an idempotent. A ring is called biregular if every principal ideal is generated by a central idempotent. In this paper, we identify the ideals and principal ideals, the annihilators of ideals, and the central and semi-central idempotents of a generalized [Formula: see text] matrix ring. We characterize the generalized matrix rings that are quasi-Baer, right p.q.-Baer, prime, and biregular. We provide examples to illustrate these concepts.

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