scholarly journals Algebraic Structure of Supernilpotent Radical Class Constructed from a Topology Thychonoff Space

2020 ◽  
Vol 11 (2) ◽  
pp. 331-338
Author(s):  
Puguh Wahyu Prasetyo ◽  
Dian Ariesta Yuwaningsih ◽  
Burhanudin Arif Nurnugroho
Keyword(s):  
Positive Integer ◽  
Algebraic Structure ◽  
Matrix Ring ◽  
Radical Class ◽  
Tychonoff Space ◽  
The Matrix ◽  
Isolated Point

A radical class of rings is called a supernilpotent radicals if it is hereditary and it contains the class  for some positive integer  In this paper, we start by exploring the concept of Tychonoff space to build a supernilpotent radical. Let  be a Tychonoff space that does not contain any isolated point. The set  of all continuous real-valued functions defined on  is a prime essential ring. Finally, we can show that the class  of rings is a supernilpotent radical class containing the matrix ring .  

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].

Hamiltonian trace graph of matrices

2021 ◽  
pp. 2250001
Author(s):  
M. Sivagami ◽  
T. Tamizh Chelvam
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Undirected Graph ◽  
Matrix Ring ◽  
Ideal Formula ◽  
The Matrix ◽  
Vertex Set ◽  
Matrix Formula ◽  
The Ideal

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be a positive integer and [Formula: see text] be the set of all [Formula: see text] matrices over [Formula: see text] For a matrix [Formula: see text] Tr[Formula: see text] is the trace of [Formula: see text] The trace graph of the matrix ring [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text][Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] The ideal-based trace graph of the matrix ring [Formula: see text] with respect to an ideal [Formula: see text] of [Formula: see text] denoted by [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if Tr[Formula: see text] In this paper, we investigate some properties and structure of [Formula: see text] Further, it is proved that both [Formula: see text] and [Formula: see text] are Hamiltonian.

Wear Model for Piston Ring and Cylinder Bore System

2005 ◽  
Author(s):  
Yunchao Qiu ◽  
Qian Zou ◽  
Gary C. Barber ◽  
Harold E. McCormick ◽  
Dequan Zou ◽  
...  
Keyword(s):  
Piston Ring ◽  
Thickness Distribution ◽  
Matrix Ring ◽  
Experimental Investigations ◽  
General Tendency ◽  
Wear Model ◽  
Wear Process ◽  
The Matrix ◽  
Surface Profiles ◽  
Cylinder Bore

A new wear model for piston ring and cylinder bore system has been developed to predict wear process with high accuracy and efficiency. It will save time and cost compared with experimental investigations. Surfaces of ring and bore were divided into small domains and assigned to corresponding elements in two-dimensional matrix. Fast Fourier Transform (FFT) and Conjugate Gradient Method (CGM) were applied to obtain pressure distribution on the computing domain. The pressure and film thickness distribution were provided by a previously developed ring/bore lubrication module. By changing the wear coefficients of the ring and bore with accumulated cycles, wear was calculated point by point in the matrix. Ring and bore surface profiles were modified when wear occurred. The results of ring and bore wear after 1 cycle, 10 cycles and 2 hours at 3600 rpm were calculated. They coincided well with the general tendency of wear in a ring and bore system.

scholarly journals QUANTUM GROUP APPROACH TO A SOLUBLE VERTEX MODEL WITH GENERALIZED ICE RULE

1996 ◽  
Vol 11 (10) ◽  
pp. 1747-1761
Author(s):  
C.L. SOW ◽  
T.T. TRUONG
Keyword(s):  
Free Energy ◽  
Quantum Mechanics ◽  
Positive Integer ◽  
Quantum Group ◽  
Algebraic Structure ◽  
Square Lattice ◽  
Free Fermion ◽  
Vertex Model ◽  
Group Approach ◽  
Operator Structure

Using the representation of the quantum group SL q(2) by the Weyl operators of the canonical commutation relations in quantum mechanics, we construct and solve a new vertex model on a square lattice. Random variables on horizontal bonds are Ising variables, and those on the vertical bonds take half positive integer values. The vertex is subjected to a generalized form of the so-called “ice rule,” its property is studied in detail and its free energy calculated with the method of quantum inverse scattering. Remarkably, in analogy with the usual six-vertex model, there exists a “free-fermion” limit with a novel rich operator structure. The existing algebraic structure suggests a possible connection with a lattice neutral plasma of charges, via the fermion-boson correspondence.

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 The normalizer of Γ0(N) in PSL(2, ℝ)

1990 ◽  
Vol 32 (3) ◽  
pp. 317-327 ◽  
Author(s):  
M. Akbas ◽  
D. Singerman
Keyword(s):  
Positive Integer ◽  
Modular Group ◽  
Prime Divisors ◽  
Möbius Transformations ◽  
The Matrix ◽  
Image Position ◽  
Mobius Transformations

Let Γ denote the modular group, consisting of the Möbius transformationsAs usual we denote the above transformation by the matrix remembering that V and – V represent the same transformation. If N is a positive integer we let Γ0(N) denote the transformations for which c ≡ 0 mod N. Then Γ0(N) is a subgroup of indexthe product being taken over all prime divisors of N.

scholarly journals Non-associative algebras containing the matrix ring

2008 ◽  
Vol 429 (1) ◽  
pp. 72-78 ◽  
Author(s):  
Jongwoo Lee ◽  
Ki-Bong Nam
Keyword(s):  
Matrix Ring ◽  
Associative Algebras ◽  
The Matrix

Subsemigroups of Stone-Čech compactifications

1994 ◽  
Vol 116 (1) ◽  
pp. 99-118 ◽  
Author(s):  
Talin Budak ◽  
Nilgün Işik ◽  
John Pym
Keyword(s):  
Positive Integer ◽  
Topological Space ◽  
Algebraic Structure ◽  
Discrete Group ◽  
Continuous Homomorphism ◽  
Discrete Space ◽  
Publication Date ◽  
Finite Subgroups ◽  
Discrete Subspace

The Stone–Čech compactification βℕ of the discrete space ℕ of positive integer is a very large topological space; for example, any countable discrete subspace of the growth ℕ* = βℕ/ℕ has a closure which is homeomorphic to βℕ itself ([23], §3·5] Now ℕ, while hardly inspiring as a discrete topological space, has a rich algebrai structure. That βℕ also has a semigroup structure which extends that of (ℕ, +) and in which multiplication is continuous in one variable has been apparent for about 30 years. (Civin and Yood [3] showed that βG was a semigroup for each discrete group G, and any mathematician could then have spotted that βℕ was a subsemigroup of βℕ.) The question which now appears natural was explicitly raised by van Douwen[6] in 1978 (in spite of the recent publication date of his paper), namely, does ℕ* contain subspaces simultaneously algebraically isomorphic and homeomorphic to βℕ? Progress on this question was slight until Strauss [22] solved it in a spectacular fashion: the image of any continuous homomorphism from βℕ into ℕ* must be finite, and so the homomorphism cannot be injective. This dramatic advance is not the end of the story. It is still not known whether that image can contain more than one point. Indeed, what appears to be one of the most difficult questions about the algebraic structure of βℕ is whether it contains any non-trivial finite subgroups

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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