Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

On nilpotent elements of ore extensions

2017 ◽  
Vol 10 (03) ◽  
pp. 1750043
Author(s):  
Masoud Azimi ◽  
Ahmad Moussavi
Keyword(s):  
Power Series ◽  
Associative Ring ◽  
General Setting ◽  
Nilpotent Elements ◽  
Ore Extensions ◽  
Ring Extensions ◽  
Series Type

Let [Formula: see text] be an associative ring with unity, [Formula: see text] be an endomorphism of [Formula: see text] and [Formula: see text] an [Formula: see text]-derivation of [Formula: see text]. We introduce the notion of [Formula: see text]-nilpotent p.p.-rings, and prove that the [Formula: see text]-nilpotent p.p.-condition extends to various ring extensions. Among other results, we show that, when [Formula: see text] is a nil-[Formula: see text]-compatible and [Formula: see text]-primal ring with [Formula: see text] nilpotent, then [Formula: see text]; and when [Formula: see text] is a nil Armendriz ring of skew power series type with [Formula: see text] nilpotent, then [Formula: see text] where [Formula: see text] is the set of nilpotent elements of [Formula: see text]. These results extend existing results to a more general setting.

scholarly journals On n-generalized commutators and Lie ideals of rings

2021 ◽  
pp. 2250221
Author(s):  
Peter V. Danchev ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Classical Result ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Simple Ring ◽  
Noncommutative Polynomials ◽  
Generalized Commutator

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

scholarly journals The radius of unit graphs of rings

2021 ◽  
Vol 6 (10) ◽  
pp. 11508-11515
Author(s):  
Zhiqun Li ◽  
◽  
Huadong Su
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Unit Graph ◽  
Vertex Set ◽  
Ring Extensions ◽  
Nonzero Identity

<abstract><p>Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.</p></abstract>

scholarly journals On the mappingxy→(xy)nin an associative ring

2004 ◽  
Vol 2004 (26) ◽  
pp. 1393-1396
Author(s):  
Scott J. Beslin ◽  
Awad Iskander
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Group Homomorphism ◽  
Artinian Rings ◽  
Commutativity Theorems

We consider the following condition (*) on an associative ringR:(*). There exists a functionffromRintoRsuch thatfis a group homomorphism of(R,+),fis injective onR2, andf(xy)=(xy)n(x,y)for some positive integern(x,y)>1. Commutativity and structure are established for Artinian ringsRsatisfying (*), and a counterexample is given for non-Artinian rings. The results generalize commutativity theorems found elsewhere. The casen(x,y)=2is examined in detail.

scholarly journals A class of rings which are algebric over the integers

1979 ◽  
Vol 2 (4) ◽  
pp. 627-650 ◽  
Author(s):  
Douglas F. Rall
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Structure Theory ◽  
Division Algebras ◽  
Periodic Rings ◽  
Prime Fields ◽  
Left And Right

A well-known theorem of N. Jacobson states that any periodic associative ring is commutative. Several authors (most notably Kaplansky and Herstein) generalized the “periodic polynomial” condition and were still able to conclude that the rings under consideration were commutative. (See [3]) In this paper we develop a structure theory for a class of rings which properly contains the periodic rings. In particular, an associative ringRis said to be a quasi-anti-integral (QAI) ring if for everya≠0inRthere exist a positive integerkand integersn1,n2,…,nk(all depending ona), so that0≠n1a=n2a2+…+nkak. In the main theorems of this paper, we show that any QAl-ring is a subdirect sum of prime QAl-rings, which in turn are shown to be left and right orders in division algebras which are algebraic over their prime fields.

scholarly journals On uniform bounds of primeness in matrix rings

2004 ◽  
Vol 76 (2) ◽  
pp. 167-174 ◽  
Author(s):  
Konstantin I. Beidar ◽  
Robert Wisbauer
Keyword(s):  
Positive Integer ◽  
Division Algebra ◽  
Associative Ring ◽  
Matrix Ring ◽  
Maximal Dimension ◽  
Uniform Bounds ◽  
Matrix Rings ◽  
The Matrix ◽  
Rank One

AbstractA subset S of an associative ring R is a uniform insulator for R provided a S b ≠ 0 for any nonzero a, b ∈ R. The ring R is called uniformly strongly prime of bound m if R has uniform insulators and the smallest of those has cardinality m. Here we compute these bounds for matrix rings over fields and obtain refinements of some results of van den Berg in this context.More precisely, for a field F and a positive integer k, let m be the bound of the matrix ring Mk(F), and let n be dimF(V), where V is a subspace of Mk(F) of maximal dimension with respect to not containing rank one matrices. We show that m + n = k2. This implies, for example, that n = k2 − k if and only if there exists a (nonassociative) division algebra over F of dimension k.

A Note on GP-Injectivity

2009 ◽  
Vol 16 (04) ◽  
pp. 625-629
Author(s):  
Yasuyuki Hirano ◽  
Hong Kee Kim ◽  
Jin Yong Kim
Keyword(s):  
Positive Integer ◽  
Left Ideal ◽  
Nonzero Element ◽  
Positive Answer

New characteristic properties of left GPGV-rings are given. It is shown that if R is a left GPGV-ring, then for any nonzero element a in R, there is a positive integer n such that an≠ 0 and (RaR+ l(an))⊕ L=R for some left ideal L contained in Soc (RR). As a corollary of this result, we are able to give a positive answer to a question raised by Yue Chi Ming.

scholarly journals On the Power Map and Ring Commutativity

1978 ◽  
Vol 21 (4) ◽  
pp. 399-404 ◽  
Author(s):  
Howard E. Bell
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Commutator Ideal ◽  
Image Position

Let R denote an associative ring with 1, let n be a positive integer, and let k = 1, 2, or 3. The ring R will be called an (n, k)-ring if it satisfies the identitiesfor all integers m with n ≤ m ≤ n + k - 1. It was shown years ago by Herstein (See [2], [9], and [10]) that for n >1, any (n, l)-ring must have nil commutator ideal C(R). Later Luh [12] proved that primary (rc, 3)-rings must in fact be commutative, and Ligh and Richoux [11] recently showed that all (n, 3)-rings are commutative.

scholarly journals Commutativity results for rings

1988 ◽  
Vol 38 (2) ◽  
pp. 191-195 ◽  
Author(s):  
Hazar Abu-Khuzam
Keyword(s):  
Positive Integer ◽  
Finite Subset ◽  
Associative Ring ◽  
Free Ring ◽  
Commutator Ideal ◽  
Fixed Positive Integer ◽  
Torsion Free

Let R be an associative ring. We prove that if for each finite subset F of R there exists a positive integer n = n(F) such that (xy)n − yn xn is in the centre of R for every x, y in F, then the commutator ideal of R is nil. We also prove that if n is a fixed positive integer and R is an n(n + 1)-torsion-free ring with identity such that (xy)n − ynxn = (yx)n xnyn is in the centre of R for all x, y in R, then R is commutative.

On the commutativity of some class of rings

1976 ◽  
Vol 21 (3) ◽  
pp. 376-380
Author(s):  
Abdullah Harmanci
Keyword(s):  
Positive Integer ◽  
Integral Domain ◽  
Associative Ring ◽  
Zero Divisors ◽  
Engel Condition ◽  
Nontrivial Zero

Throughout, R will denote an associative ring with center Z. For elements x, y of R and k a positive integer, we define inductively [x, y]0 = x, [x, y] = [x, y]1 = xy − yx, [x, y, y, hellip, y]k = [[x, y, y, hellip, y]k−1, y]. A ring R is said to satisfy the k-th Engel condition if [x, y, y, hellip, y]k = 0. By an integral domain we mean a nonzero ring without nontrivial zero divisors.

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