scholarly journals A note on uniform bounds of primeness in matrix rings

1998 ◽  
Vol 65 (2) ◽  
pp. 212-223 ◽  
Author(s):  
John E. Van den Berg
Keyword(s):  
Positive Integer ◽  
Matrix Ring ◽  
Uniform Bound ◽  
Prime Rings ◽  
Element Subset ◽  
Uniform Bounds ◽  
Matrix Rings ◽  
Division Rings ◽  
Nonassociative Algebras

AbstractA nonzero ring R is said to be uniformly strongly prime (of bound n) if n is the smallest positive integer such that for some n-element subset X of R we have xXy ≠ 0 whenever 0 ≠ x, y ∈ R. The study of uniformly strongly prime rings reduces to that of orders in matrix rings over division rings, except in the case n = 1. This paper is devoted primarily to an investigation of uniform bounds of primeness in matrix rings over fields. It is shown that the existence of certain n-dimensional nonassociative algebras over a field F decides the uniform bound of the n × n matrix ring over F.

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.

Representation of tiled matrix rings as full matrix rings

1989 ◽  
Vol 105 (1) ◽  
pp. 67-72 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Positive Integer ◽  
Jacobson Radical ◽  
Integral Domain ◽  
Matrix Ring ◽  
The Other ◽  
Matrix Rings ◽  
Full Matrix ◽  
Image Position ◽  
Full Matrix Ring ◽  
Unique Maximal Ideal

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and setThen T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

Rings with fine idempotents

2020 ◽  
pp. 2250013
Author(s):  
Grigore Călugăreanu ◽  
Yiqiang Zhou
Keyword(s):  
Positive Integer ◽  
Matrix Ring ◽  
Matrix Rings ◽  
New Class ◽  
Unital Ring ◽  
The Matrix

An idempotent in a ring is called fine (see G. Călugăreanu and T. Y. Lam, Fine rings: A new class of simple rings, J. Algebra Appl. 15(9) (2016) 18) if it is a sum of a nilpotent and a unit. A ring is called an idempotent-fine ring (briefly, an [Formula: see text] ring) if all its nonzero idempotents are fine. In this paper, the properties of [Formula: see text] rings are studied. A notable result is proved: The diagonal idempotents [Formula: see text] ([Formula: see text]) are fine in the matrix ring [Formula: see text] for any unital ring [Formula: see text] and any positive integer [Formula: see text]. This yields many classes of rings over which matrix rings are [Formula: see text].

Semiprime Goldie Generalised Matrix Rings

1995 ◽  
Vol 38 (2) ◽  
pp. 174-176 ◽  
Author(s):  
Michael V. Clase
Keyword(s):  
Matrix Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Rings ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions are given for a generalised matrix ring to be semiprime right Goldie.

scholarly journals Carbonitration of a Tool for Pressing Stainless Steel Pipes

2020 ◽  
Vol 7 (2) ◽  
pp. C17-C21
Author(s):  
I. V. Ivanov ◽  
M. V. Mohylenets ◽  
K. A. Dumenko ◽  
L. Kryvchyk ◽  
T. S. Khokhlova ◽  
...  
Keyword(s):  
Heat Treatment ◽  
Thermal Treatment ◽  
Heat Resistance ◽  
Alkali Metals ◽  
Matrix Ring ◽  
High Hardness ◽  
Operating Conditions ◽  
Matrix Rings ◽  
Conventional Heat Treatment ◽  
The Stability

To upgrade the operational stability of the tool at LLC “Karbaz”, Sumy, Ukraine, carbonation of tools and samples for research in melts of salts of cyanates and carbonates of alkali metals at 570–580 °C was carried out to obtain a layer thickness of 0.15–0.25 mm and a hardness of 1000–1150 НV. Tests of the tool in real operating conditions were carried out at the press station at LLC “VO Oscar”, Dnipro, Ukraine. The purpose of the test is to evaluate the feasibility of carbonitriding of thermo-strengthened matrix rings and needle-mandrels to improve their stability, hardness, heat resistance, and endurance. If the stability of matrix rings after conventional heat setting varies around 4–6 presses, the rings additionally subjected to chemical-thermal treatment (carbonitration) demonstrated the stability of 7–9 presses due to higher hardness, heat resistance, the formation of a special structure on the surface due to carbonitration in salt melts cyanates and carbonates. Nitrogen and carbon present in the carbonitrided layer slowed down the processes of transformation of solid solutions and coagulation of carbonitride phases. The high hardness of the carbonitrified layer is maintained up to temperatures above 650 °C. If the stability of the needle-mandrels after conventional heat treatment varies between 50–80 presses, the needles, additionally subjected to chemical-thermal treatment (carbonitration) showed the stability of 100–130 presses due to higher hardness, wear resistance, heat resistance, the formation of a special surface structure due to carbonitration in melts of salts of cyanates and carbonates. Keywords: needle-mandrel, matrix ring, pressing, heat treatment, carbonitration.

Derivations of Higher Order in Prime Rings

1996 ◽  
Vol 39 (3) ◽  
pp. 376-384 ◽  
Author(s):  
Youpei Ye ◽  
Jiang Luh
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Higher Order ◽  
Prime Rings

AbstractLet R be a prime ring of characteristic not 2 and d a derivation of R. It is shown that if d2n is a derivation of R, where n is a positive integer, then d2n-1 = 0.

On semireversible rings

2019 ◽  
pp. 2150018
Author(s):  
Debraj Roy ◽  
Tikaram Subedi
Keyword(s):  
Positive Integer ◽  
Matrix Rings ◽  
New Class ◽  
Ring Of Fractions ◽  
Strongly Regular

In this paper, we introduce and study a new class of rings which we call semireversible rings. A ring [Formula: see text] is called semireversible if for any [Formula: see text] implies there exists a positive integer [Formula: see text] such that [Formula: see text]. We observe that the class of semireversible rings strictly lies between the class of central reversible rings and weakly reversible rings. Some relations are provided between semireversible rings and many other known classes of rings. Some extensions of semireversible rings such as ring of fractions, Dorroh extension, subrings of matrix rings are investigated. Finally, we study semireversible rings via modules over them wherein among other results, we prove that a semireversible left (right) SF-ring is strongly regular.

scholarly journals Complementation of the Jacobson group in a matrix ring

2005 ◽  
Vol 72 (2) ◽  
pp. 317-324
Author(s):  
David Dolžan
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Matrix Ring ◽  
Graded Rings ◽  
Matrix Rings ◽  
Generators And Relations ◽  
Group Of Units ◽  
The Matrix ◽  
Finite Commutative Ring

The Jacobson group of a ring R (denoted by  = (R)) is the normal subgroup of the group of units of R (denoted by G(R)) obtained by adding 1 to the Jacobson radical of R (J(R)). Coleman and Easdown in 2000 showed that the Jacobson group is complemented in the group of units of any finite commutative ring and also in the group of units a n × n matrix ring over integers modulo ps, when n = 2 and p = 2, 3, but it is not complemented when p ≥ 5. In 2004 Wilcox showed that the answer is positive also for n = 3 and p = 2, and negative in all the remaining cases. In this paper we offer a different proof for Wilcox's results and also generalise the results to a matrix ring over an arbitrary finite commutative ring. We show this by studying the generators and relations that define a matrix ring over a field. We then proceed to examine the complementation of the Jacobson group in the matrix rings over graded rings and prove that complementation depends only on the 0-th grade.

EXTREMAL CAYLEY DIGRAPHS OF FINITE ABELIAN GROUPS

2011 ◽  
Vol 12 (01n02) ◽  
pp. 125-135 ◽  
Author(s):  
ABBY GAIL MASK ◽  
JONI SCHNEIDER ◽  
XINGDE JIA
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Communication Networks ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Element Subset ◽  
Cayley Digraph ◽  
Positive Integers ◽  
Element Set

Cayley digraphs of finite abelian groups are often used to model communication networks. Because of their applications, extremal Cayley digraphs have been studied extensively in recent years. Given any positive integers d and k. Let m*(d, k) denote the largest positive integer m such that there exists an m-element finite abelian group Γ and a k-element subset A of Γ such that diam ( Cay (Γ, A)) ≤ d, where diam ( Cay (Γ, A)) denotes the diameter of the Cayley digraph Cay (Γ, A) of Γ generated by A. Similarly, let m(d, k) denote the largest positive integer m such that there exists a k-element set A of integers with diam (ℤm, A)) ≤ d. In this paper, we prove, among other results, that [Formula: see text] for all d ≥ 1 and k ≥ 1. This means that the finite abelian group whose Cayley digraph is optimal with respect to its diameter and degree can be a cyclic group.

Derivations of some classes of matrix rings

2017 ◽  
Vol 16 (02) ◽  
pp. 1750027 ◽  
Author(s):  
Feride Kuzucuoğlu ◽  
Umut Sayın
Keyword(s):  
Associative Ring ◽  
Matrix Ring ◽  
Matrix Rings ◽  
Ideal Formula ◽  
The Matrix

Let [Formula: see text] be the ring of all (lower) niltriangular [Formula: see text] matrices over any associative ring [Formula: see text] with identity and [Formula: see text] be the ring of all [Formula: see text] matrices over an ideal [Formula: see text] of [Formula: see text]. We describe all derivations of the matrix ring [Formula: see text].

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