• Korean Journal of Mathematics
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• 제21권4호
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• pp.463-471
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• 2013

Let R be a ring with identity 1, $I(R){\neq}\{0\}$ be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, $f{\in}I(R)$ ($e{\neq}f$), $e+f{\in}I(R)$. In this paper, the following are shown: (1) I(R) is a finite additive set if and only if $M(R){\backslash}\{0\}$ is a complete set of primitive central idempotents, char(R) = 2 and every nonzero idempotent of R can be expressed as a sum of orthogonal primitive idempotents of R; (2) for a regular ring R such that I(R) is a finite additive set, if the multiplicative group of all units of R is abelian (resp. cyclic), then R is a commutative ring (resp. R is a finite direct product of finite field).

• 대한수학회지
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• 제46권2호
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• pp.249-256
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• 2009

Let $G{\otimes}G$ be the tensor square of a group G. The set of all elements a in G such that $a{\otimes}g\;=\;1_{\otimes}$, for all g in G, is called the tensor centre of G and denoted by $Z^{\otimes}^$(G). In this paper some properties of the tensor centre of G are obtained and the capability of the pair of groups (G, G') is determined. Finally, the structure of $J_2$(G) will be described, where $J_2$(G) is the kernel of the map $\kappa$ : $G{\otimes}\;{\rightarrow}\;G'$.

• 대한수학회지
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• 제40권2호
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• pp.225-239
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• 2003

(G, <) is an ordered group if'<'is a total order relation on G in which f < g implies that xfy < xgy for all f, g, x, y $\in$ G. We say that (G, <) is centrally ordered if (G, <) is ordered and ［G,D］ $\subseteq$ C for every convex jump C $\prec$ D in G. Equivalently, if $f^{-1}g f{\leq} g^2$ for all f, g $\in$ G with g > 1. Every order on a torsion-free locally nilpotent group is central. We prove that if every order on every two-generator subgroup of a locally soluble orderable group G is central, then G is locally nilpotent. We also provide an example of a non-nilpotent two-generator metabelian orderable group in which all orders are central.

• 대한수학회보
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• 제37권2호
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• pp.217-227
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• 2000

This paper is motivated by the results in [2], [10], [13] and [19]. We study some properties of generalizations of commutative rings and relations between them. We also show that for a right quasi-duo right weakly ${\pi}-regular$ ring R, R is an (S,2)-ring if and only if every idempotent in R is a sum of two units in R, which gives a generalization of [2, Theorem 4] on right quasi-duo rings. Moreover we find a condition which is equivalent to the strongly ${\pi}-regularity$ of an abelian right quasi-duo ring.

• 대한수학회지
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• 제43권6호
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• pp.1269-1287
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• 2006

As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle.

• 대한수학회논문집
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• 제33권4호
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• pp.1075-1082
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• 2018

Let R be a commutative ring, G be an Abelian group, and let RG be the group ring. We say that RG is a U-group ring if a is a unit in RG if and only if ${\epsilon}(a)$ is a unit in R. We show that RG is a U-group ring if and only if G is a p-group and $p{\in}J(R)$. We give some properties of U-group rings and investigate some properties of well known rings, such as Hermite rings and rings with stable range, in the presence of U-group rings.

• 대한수학회보
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• 제57권1호
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• pp.109-115
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• 2020

Recently, Lesieutre constructed a 6-dimensional projective variety X over any field of characteristic zero whose automorphism group Aut(X) is discrete but not finitely generated. As an application, he also showed that X is an example of a projective variety with infinitely many non-isomorphic real structures. On the other hand, there are also several finiteness results of real structures of projective varieties. The aim of this short paper is to give a sufficient condition for the finiteness of real structures on a projective manifold in terms of the structure of the automorphism group. To be more precise, in this paper we show that, when X is a projective manifold of any dimension≥ 2, if Aut(X) does not contain a subgroup isomorphic to the non-abelian free group ℤ ∗ ℤ, then there are only finitely many real structures on X, up to ℝ-isomorphisms.

• Journal of applied mathematics & informatics
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• 제12권1_2호
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• pp.143-154
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• 2003

A finite group G is called (l, m, n)-generated, if it is a quotient group of the triangle group T(l, m, n) = 〈$\chi$, y, z│$\chi$$\^$l/ = y$\^$m/ = z$^n$ = $\chi$yz = 1〉. In [19], the question of finding all triples (l, m, n) such that non-abelian finite simple group are (l, m, n)-generated was posed. In this paper we partially answer this question for the sporadic group Ru. In fact, we prove that if p, q and r are prime divisors of │Ru│, where p < q < r and$.$(p, q) $\neq$ (2, 3), then Ru is (p, q, r)-generated.

• Kyungpook Mathematical Journal
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• 제47권1호
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• pp.21-30
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• 2007

We say a module $M_R$ a semicommutative module if for any $m{\in}M$ and any $a{\in}R$, $ma=0$ implies $mRa=0$. This paper gives various properties of reduced, Armendariz, Baer, Quasi-Baer, p.p. and p.q.-Baer rings to extend to modules. In addition we also prove, for a p.p.-ring R, R is semicommutative iff R is Armendariz. Let R be an abelian ring and $M_R$ be a p.p.-module, then $M_R$ is a semicommutative module iff $M_R$ is an Armendariz module. For any ring R, R is semicommutative iff A(R, ${\alpha}$) is semicommutative. Let R be a reduced ring, it is shown that for number $n{\geq}4$ and $k=[n=2]$, $T^k_n(R)$ is semicommutative ring but $T^{k-1}_n(R)$ is not.

• 한국진공학회:학술대회논문집
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• 한국진공학회 2011년도 제40회 동계학술대회 초록집
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• pp.11-11
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• 2011

In this talk, I will discuss roles of pseudo vector and scalar potential in changing physical properties of graphene systems. First, graphene under small uniaxial strain is shown to be described by the generalized Weyl's Hamiltonian with inclusion of pseudo vector and scalar potential simultaneously [1]. Thus, strained graphene is predicted to exhibit velocity anisotropy as well as work function enhancement without any gap. Second, if homogeneous strains with different strengths are applied to each layer of bilayer graphene, transverse electric fields across the two layers can be generated without any external electronic sources, thereby opening an energy gap [2]. This phenomenon is made possible by generation of inequivalent pseudo scalar potentials in the two graphene layers. Third, when very tiny lateral interlayer shift occurs in bilayer graphene, the Fermi surfaces of the system are shown to undergo Lifshitz transition [3]. We will show that this unexpected hypersensitive electronic topological transition is caused by a unique interplay between the effective non-Abelian vector potential generated by sliding motions and Berry's phases associated with massless Dirac electrons.