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.

scholarly journals On power sums of matrices over a finite commutative ring

2017 ◽  
Vol 27 (05) ◽  
pp. 547-560 ◽  
Author(s):  
P. Fortuny ◽  
J. M. Grau ◽  
A. M. Oller-Marcén ◽  
I. F. Rúa
Keyword(s):  
Commutative Ring ◽  
Matrix Ring ◽  
Power Sums ◽  
The Matrix ◽  
Finite Commutative Ring

In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].

The trace graph of the matrix ring over a finite commutative ring

2018 ◽  
Vol 156 (1) ◽  
pp. 132-144 ◽  
Author(s):  
F. A. A. Almahdi ◽  
K. Louartiti ◽  
M. Tamekkante
Keyword(s):  
Commutative Ring ◽  
Matrix Ring ◽  
The Matrix ◽  
Finite Commutative Ring

scholarly journals Complementation in the group of units of matrix rings

2004 ◽  
Vol 70 (2) ◽  
pp. 223-227 ◽  
Author(s):  
Stewart Wilcox
Keyword(s):  
Normal Subgroup ◽  
Jacobson Radical ◽  
Matrix Rings ◽  
Group Of Units

Let R be a ring with 1 and (R) its Jacobson radical. Then 1 + (R) is a normal subgroup of the group of units, G(R). The existence of a complement to this subgroup was explored in a paper by Coleman and Easdown; in particular the ring R = Matn(ℤpk) was considered. We prove the remaining cases to determine for which n, P and k a complement exists in this ring.

scholarly journals On the group of unit-valued polynomial functions

2021 ◽  
Author(s):  
Amr Ali Al-Maktry
Keyword(s):  
Commutative Ring ◽  
Semidirect Product ◽  
Polynomial Functions ◽  
Dual Numbers ◽  
Ring Of Integers ◽  
Group Of Units ◽  
Canonical Representations ◽  
Finite Commutative Ring

AbstractLet R be a finite commutative ring. The set $${{\mathcal{F}}}(R)$$ F ( R ) of polynomial functions on R is a finite commutative ring with pointwise operations. Its group of units $${{\mathcal{F}}}(R)^\times $$ F ( R ) × is just the set of all unit-valued polynomial functions. We investigate polynomial permutations on $$R[x]/(x^2)=R[\alpha ]$$ R [ x ] / ( x 2 ) = R [ α ] , the ring of dual numbers over R, and show that the group $${\mathcal{P}}_{R}(R[\alpha ])$$ P R ( R [ α ] ) , consisting of those polynomial permutations of $$R[\alpha ]$$ R [ α ] represented by polynomials in R[x], is embedded in a semidirect product of $${{\mathcal{F}}}(R)^\times $$ F ( R ) × by the group $${\mathcal{P}}(R)$$ P ( R ) of polynomial permutations on R. In particular, when $$R={\mathbb{F}}_q$$ R = F q , we prove that $${\mathcal{P}}_{{\mathbb{F}}_q}({\mathbb{F}}_q[\alpha ])\cong {\mathcal{P}}({\mathbb{F}}_q) \ltimes _\theta {{\mathcal{F}}}({\mathbb{F}}_q)^\times $$ P F q ( F q [ α ] ) ≅ P ( F q ) ⋉ θ F ( F q ) × . Furthermore, we count unit-valued polynomial functions on the ring of integers modulo $${p^n}$$ p n and obtain canonical representations for these functions.

When is the Jacobson graph projective?

2018 ◽  
Vol 17 (09) ◽  
pp. 1850168
Author(s):  
Atossa Parsapour ◽  
Khadijeh Ahmadjavaheri
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the Jacobson radical of [Formula: see text]. The Jacobson graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertex-set [Formula: see text], such that two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] is not a unit of [Formula: see text]. The goal in this paper is to list every finite commutative ring [Formula: see text] with nonzero identity (up to isomorphism) such that the graph [Formula: see text] is projective.

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].

The Metric Dimension of the Total Graph of a Finite Commutative Ring

2016 ◽  
Vol 59 (4) ◽  
pp. 748-759 ◽  
Author(s):  
David Dolžan
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Metric Dimension ◽  
Total Graph ◽  
Finite Commutative Ring

AbstractWe study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial, and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields.

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).

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.

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].

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