• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.511-519
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• 2009

A matrix $P{\in}\mathbb{C}^{n{\times}n}$ is called a generalized reflection matrix if $P^*$ = P and $P^2$ = I. An $n{\times}n$ complex matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = PAP (A = -PAP). It is well-known that the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P have many special properties and widely used in engineering and scientific computations. In this paper, we give new necessary and sufficient conditions for the existence of the reflexive (anti-reflexive) solutions to the linear matrix equation AXB + CY D = E and derive representation of the general reflexive (anti-reflexive) solutions to this matrix equation. By using the obtained results, we investigate the reflexive (anti-reflexive) solutions of some special cases of this matrix equation.

• East Asian mathematical journal
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• v.34 no.3
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• pp.305-316
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• 2018

Let R be a ring with identity. A right ideal ideal I of a ring R is called ref lexive (resp. completely ref lexive) if $aRb{\subseteq}I$ implies that $bRa{\subseteq}I$ (resp. if $ab{\subseteq}I$ implies that $ba{\subseteq}I$) for any $a,\;b{\in}R$. R is called ref lexive (resp. completely ref lexive) if the zero ideal of R is a reflexive ideal (resp. a completely reflexive ideal). Let K(R) (called the ref lexive radical of R) be the intersection of all reflexive ideals of R. In this paper, the following are investigated: (1) Some equivalent conditions on an reflexive ideal of a ring are obtained; (2) reflexive (resp. completely reflexive) property is Morita invariant; (3) For any ring R, we have $K(M_n(R))=M_n(K(R))$ where $M_n(R)$ is the ring of all n by n matrices over R; (4) For a ring R, we have $K(R)[x]{\subseteq}K(R[x])$; in particular, if R is quasi-Armendaritz, then R is reflexive if and only if R[x] is reflexive.

• Korean Journal of Mathematics
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• v.22 no.2
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• pp.217-234
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• 2014

Mason extended the reflexive property for subgroups to right ideals, and examined various connections between these and related concepts. A ring was usually called reflexive if the zero ideal satisfies the reflexive property. We here study this property skewed by ring endomorphisms, introducing the concept of an ${\alpha}$-skew reflexive ring, where is an endomorphism of a given ring.

• Korean Journal of Mathematics
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• v.17 no.3
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• pp.233-236
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• 2009

We investigate in this paper some equivalent conditions for right principally quasi-Baer rings to be reflexive. Using these results we are able to prove that if R is a reflexive right principally quasi-Baer ring then R is a left principally quasi-Baer ring. In addition, for an idempotent reflexive principally quasi-Baer ring R we show that R is prime if and only if R is torsion free.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1957-1972
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• 2013

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. In this note we study the structure of idempotents satisfying the reflexive property and introduce reflexive-idempotents-property (simply, RIP) as a generalization. It is proved that the RIP can go up to polynomial rings, power series rings, and Dorroh extensions. The structure of non-Abelian RIP rings of minimal order (with or without identity) is completely investigated.

• Kyungpook Mathematical Journal
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• v.54 no.2
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• pp.263-269
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• 2014

As a further study on reflexive families of subsets, we introduce the reflexive index for a family of subsets of a given set and show that the index of a finite family of subsets of a finite or countably infinite set is always finite. The reflexive indices of some special families are also considered.

• Journal of Fashion Business
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• v.10 no.3
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• pp.129-142
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• 2006

This research chose 135 environmental cleaners in Busan Metropolitan City as subjects. First, the reality of wearing their safety jackets was investigated. Next, visual evaluation of the reflexive function of the current safety jackets was made. Then, based on the above results and the subsequent improvement proposal, the final product, the research costume, was made. 1. Based on the investigation of the wearing reality of safety jackets and the visual evaluation results of the reflexive function, an improvement proposal was made. As cleaners are usually seen sidewise on the roads and lateral luminosity is rather low, this aspect was compensated on the part of a jacket. Also, the reflexive belt used the diagram of ${\Delta}$ to improve visual effects. 2. As the lower body held the accident frequency of 45.0%, the sides of pants had the reflexive belt to increase visibility. 3. In consideration of the mutual effects between the background cloth and general reflexive width, the same width (5cm) of the reflexive belt as that of Type A was adopted.

• Kyungpook Mathematical Journal
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• v.46 no.4
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• pp.597-601
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• 2006

We introduce in this paper the concept of idempotent reflexive right ideals and concern with rings containing an injective maximal right ideal. Some known results for reflexive rings and right HI-rings can be extended to idempotent reflexive rings. As applications, we are able to give a new characterization of regular right self-injective rings with nonzero socle and extend a known result for right weakly regular rings.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.125-135
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• 2015

Let R be a commutative ring and U an R-module. The aim of this paper is to study the duality between U-reflexive (pre)envelopes and U-reflexive (pre)covers of R-modules.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.81-105
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• 2018

In this paper we give two characterisations of the class of reflexive graphs admitting distributive lattice polymorphisms and use these characterisations to address the problem of recognition: we find a polynomial time algorithm to decide if a given reflexive graph G, in which no two vertices have the same neighbourhood, admits a distributive lattice polymorphism.