Some new characterizations of periodic rings

2019 ◽  
Vol 19 (12) ◽  
pp. 2050235 ◽  
Author(s):  
Jian Cui ◽  
Peter Danchev
Keyword(s):  
Periodic Ring ◽  
Periodic Rings ◽  
Positive Integers ◽  
Unital Rings ◽  
Regular Rings ◽  
Comprehensive Study

A ring [Formula: see text] is called periodic if, for every [Formula: see text] in [Formula: see text], there exist two distinct positive integers [Formula: see text] and [Formula: see text] such that [Formula: see text]. The paper is devoted to a comprehensive study of the periodicity of arbitrary unital rings. Some new characterizations of periodic rings and their relationship with strongly [Formula: see text]-regular rings are provided as well as, furthermore, an application of the obtained main results to a ∗-version of a periodic ring is being considered. Our theorems somewhat considerably improved on classical results in this direction.

scholarly journals Generalized periodic rings

1996 ◽  
Vol 19 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Howard E. Bell ◽  
Adil Yaqub
Keyword(s):  
Opposite Parity ◽  
Periodic Ring ◽  
Periodic Rings ◽  
Positive Integers

LetRbe a ring, and letNandCdenote the set of nilpotents and the center ofR, respectively.Ris called generalized periodic if for everyx∈R\(N⋃C), there exist distinct positive integersm,nof opposite parity such thatxn−xm∈N⋂C. We prove that a generalized periodic ring always has the setNof nilpotents forming an ideal inR. We also consider some conditions which imply the commutativity of a generalized periodic ring.

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 Invo-regular unital rings

2018 ◽  
Vol 72 (1) ◽  
pp. 45
Author(s):  
Peter V. Danchev
Keyword(s):  
Regular Ring ◽  
A Priori ◽  
Proper Subclass ◽  
In Trans ◽  
Unital Rings ◽  
Regular Rings

It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without any concrete indications a priori, these rings are manifestly even commutative invo-clean as defined by the author in Commun. Korean Math. Soc., 2017.

scholarly journals On periodic rings

2001 ◽  
Vol 25 (6) ◽  
pp. 417-420
Author(s):  
Xiankun Du ◽  
Qi Yi
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sum ◽  
Periodic Rings ◽  
Necessary And Sufficient ◽  
Nil Ring ◽  
Positive Integers

It is proved that a ring is periodic if and only if, for any elementsxandy, there exist positive integersk,l,m, andnwith eitherk≠morl≠n, depending onxandy, for whichxkyl=xmyn. Necessary and sufficient conditions are established for a ring to be a direct sum of a nil ring and aJ-ring.

scholarly journals Group-regular rings

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3551-3560
Author(s):  
Xavier Mary ◽  
Pedro Patrício
Keyword(s):  
Stable Range ◽  
Stable Range One ◽  
Unital Rings ◽  
Regular Rings

We propose different generalizations of unit-regularity of elements in general rings (non necessarily unital rings). We then study general rings for which all elements have these properties. We notably compare them with unit-regular ideals and general rings with stable range one. We also prove that these rings are morphic rings.

scholarly journals Some characterizations of π-regular rings with no infinite trivial subring

1991 ◽  
Vol 34 (1) ◽  
pp. 1-5
Author(s):  
Yasuyuki Hirano
Keyword(s):  
Commutative Ring ◽  
Regular Ring ◽  
Finite Ring ◽  
Periodic Ring ◽  
Strongly Regular ◽  
Regular Rings

It is shown that a ringRis a π-regular ring with no infinite trivial subring if and only ifRis a subdirect sum of a strongly regular ring and a finite ring. Some other characterizations of such a ring are given. Similar result is proved for a periodic ring. As a corollary, it is shown that every δ-ring is a subdirect sum of a Unite ring and a commutative ring. This was conjectured by Putcha and Yaqub.

Structure of Certain Periodic Rings

1985 ◽  
Vol 28 (1) ◽  
pp. 120-123
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Zero Divisor ◽  
Boolean Ring ◽  
Zero Divisors ◽  
Periodic Ring ◽  
Periodic Rings

AbstractLet R be a periodic ring, N the set of nilpotents, and D the set of right zero divisors of R. Suppose that (i) N is commutative, and (ii) every x in R can be uniquely written in the form x = e + a, where e2 = e and a ∊ N. Then N is an ideal in R and R/N is a Boolean ring. If (i) is satisfied but (ii) is now assumed to hold merely for those elements x ∊ D, and if 1 ∊ R, then N is still an ideal in R and R/N is a subdirect sum of fields. It is further shown that if (i) is satisfied but (ii) is replaced by: "every right zero divisor is either nilpotent or idempotent," and if 1 ∊ R, then N is still an ideal in R and R/N is either a Boolean ring or a field.

A Note on Periodic Rings

2008 ◽  
Vol 15 (02) ◽  
pp. 199-206
Author(s):  
H. R. Dorbidi
Keyword(s):  
Natural Number ◽  
Unit Group ◽  
Periodic Ring ◽  
Periodic Rings

Let S be a semigroup. The degree of S is the smallest natural number r such that for each x ∈ S, xn(x)+r = xn(x), where n(x) ∈ ℕ. If such a number r does not exist, we say that the degree of S is infinite. For a group G, this coincides with the exponent of G. We prove that for a periodic ring R, the degree of R equals exp (U(R)), where U(R) denotes the unit group of R. Then we determine all degrees for any rings.

Commutativity Conditions for Rings and Generalized 2-CN Rings

2021 ◽  
Vol 28 (01) ◽  
pp. 51-62
Author(s):  
Hua Yao ◽  
Jianhua Sun ◽  
Junchao Wei
Keyword(s):  
Positive Integer ◽  
Positive Integers ◽  
Regular Rings

Firstly, the commutativity of rings is investigated in this paper. Let [Formula: see text] be a ring with identity. Then we obtain the following commutativity conditions: (1) if for each [Formula: see text] and each [Formula: see text], [Formula: see text] for [Formula: see text], where [Formula: see text] and [Formula: see text] are relatively prime positive integers, then [Formula: see text] is commutative; (2) if for each [Formula: see text] and each [Formula: see text], [Formula: see text] for [Formula: see text], where [Formula: see text] is a positive integer, then [Formula: see text] is commutative. Secondly, generalized 2-CN rings, a kind of ring being commutative to some extent, are investigated. Some relations between generalized 2-CN rings and other kinds of rings, such as reduced rings, regular rings, 2-good rings, and weakly Abel rings, are presented.

scholarly journals A Commutativity Theorem for Near-Rings

1977 ◽  
Vol 20 (1) ◽  
pp. 25-28 ◽  
Author(s):  
Howard E. Bell
Keyword(s):  
Affirmative Answer ◽  
Commutativity Theorem ◽  
Nilpotent Elements ◽  
Periodic Ring ◽  
Positive Integers

A ring or near-ring R is called periodic if for each xϵR, there exist distinct positive integers n, m for which xn = xm. A well-known theorem of Herstein states that a periodic ring is commutative if its nilpotent elements are central [5], and Ligh [6] has asked whether a similar result holds for distributively-generated (d-g) near-rings. It is the purpose of this note to provide an affirmative answer.

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