• 한국정보과학회논문지:시스템및이론
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• 제27권3호
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• pp.313-323
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• 2000

본 논문에서는 상호연결망 그래프 중 하나인 (n,k)-스타 그래프에 대한 링 임베딩 문제를 다룬다. (n,k)-스타 그래프에 대한 링 임베딩 전략의 유연성을 개선한 새로운 임베딩 기법을 제시하고, 아울러 에지에 결함을 갖는 경우의 결함허용 링 임베딩 문제에 응용 가능함을 보여줌으로써 본 기법의 확장성에서의 우수함을 밝히고자 한다. 본 논문에서 다루고 있는 사이클 특성 관련 연구는 병렬처리 분야에서의 멀티캐스팅 등과 같이 내재된 사이클 특성을 활용하는 분야에 응용이 가능하다.

• 대한수학회논문집
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• 제27권1호
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• pp.57-68
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• 2012

The commuting graph of an arbitrary ring R, denoted by ${\Gamma}(R)$, is a graph whose vertices are all non-central elements of R, and two distinct vertices a and b are adjacent if and only if ab = ba. In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of group ring $Z_nQ_8$. The main result is that $\Gamma(Z_nQ_8)$ is connected if and only if n is not a prime. If $\Gamma(Z_nQ_8)$ is connected, then diam($Z_nQ_8$)= 3, while $\Gamma(Z_nQ_8)$ is disconnected then every connected component of $\Gamma(Z_nQ_8)$ must be a complete graph with a same size. Further, we obtain the degree of every vertex in $\Gamma(Z_nQ_8)$, the maximum degree and the minimum degree of $\Gamma(Z_nQ_8)$.

• Journal of applied mathematics & informatics
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• 제9권2호
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• pp.851-857
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• 2002

In this paper, we introduce a congruence relation on a seminear-ring and study quotient structures on it. Also, we investigate homomorphisms on a seminear-ring.

• Journal of applied mathematics & informatics
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• 제26권3_4호
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• pp.541-547
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• 2008

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with $n\;{\geq}\;3$. We denote by $M_n(R)$ the ring of all $n{\times}n$ matrices over R. Let ($J_n(R)$) be the additive subgroup of $M_n(R)$ generated additively by all idempotent matrices. Let ($D=J_n(R)$) or $M_n(R)$. In this paper, by using an additive idem potence-preserving result obtained by Coo (see [4]), I characterize (i) the additive preservers of tripotence from D to $M_m(R)$ when 2 and 3 are units of R; (ii) the additive preservers of inverses (respectively, Drazin inverses, group inverses, {1}-inverses, {2}-inverses, {1, 2}-inverses) from $M_n(R)$ to $M_n(R)$ when 2 and 3 are units of R.

• Bulletin of the Korean Chemical Society
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• 제26권9호
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• pp.1339-1343
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• 2005

The N-inversion energies and nucleophilic ring-opening reactions of N-substituted aziridine compounds are investigated using B3LYP/6-31+$G^*$ methods, where substituents (R) on the nitrogen atom has been H (1), Me (2), Ph (3), Bn (4), CHMePh (5), $CO_2Me$ (6), COPh (7) and $SO_2Ph$ (8). The N-inversion energy with X group are decreased as the following order: R = CHMePh (17.06 kcal/mol) $\gt$ Me (16.97) $\gt$ Bn (16.70) $\gt$ H (16.64) $\gt$ $SO_2Ph$ (12.18) $\gt$ Ph (8.91) $\gt$ COPh (5.75) $\gt$ $CO_2Me$ (5.48). For reactivity of the ring opening toward cyanide ion, the aziridine 6 (R=$CO_2Me$) is shown to be the most reactive one. During the ring opening of aziridine 6 by CN$^{\ominus}$, the torsional OCNC angle becomes near to $180^{\circ}$, where the geometry allows for the effective incorporation of electrons of the nitrogen atom to the C=O bond. It would be a possible driving force for nucleophilic ring opening reaction as well as decreasing the N-inversion energy barrier. Regarding to the regioselectivity, the orientation of nucleophile in ring opening reaction appears to be different in the case of 9 and 10. The results are discussed in terms of steric/electronic effect of the $C_2$-substituents.

• 대한수학회보
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• 제50권5호
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• pp.1433-1439
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• 2013

A ring R is called left morphic if $$R/Ra{\simeq_-}l(a)$$ for every $a{\in}R$. Equivalently, for every $a{\in}R$ there exists $b{\in}R$ such that $Ra=l(b)$ and $l(a)=Rb$. A ring R is called left quasi-morphic if there exist $b$ and $c$ in R such that $Ra=l(b)$ and $l(a)=Rc$ for every $a{\in}R$. A result of T.-K. Lee and Y. Zhou says that R is unit regular if and only if $$R[x]/(x^2){\simeq_-}R{\propto}R$$ is morphic. Motivated by this result, we investigate the morphic property of the ring $$S_n=^{def}R[x_1,x_2,{\cdots},x_n]/(\{x_ix_j\})$$, where $i,j{\in}\{1,2,{\cdots},n\}$. The morphic elements of $S_n$ are completely determined when R is strongly regular.

• 대한수학회보
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• 제47권5호
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• pp.907-913
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• 2010

Let D be an integrally closed domain with quotient field K, * be a star operation on D, X, Y be indeterminates over D, $N_*\;=\;\{f\;{\in}\;D[X]|\;(c_D(f))^*\;=\;D\}$ and $R\;=\;D[X]_{N_*}$. Let b be the b-operation on R, and let $*_c$ be the star operation on D defined by $I^{*_c}\;=\;(ID[X]_{N_*})^b\;{\cap}\;K$. Finally, let Kr(R, b) (resp., Kr(D, $*_c$)) be the Kronecker function ring of R (resp., D) with respect to Y (resp., X, Y). In this paper, we show that Kr(R, b) $\subseteq$ Kr(D, $*_c$) and Kr(R, b) is a kfr with respect to K(Y) and X in the notion of [2]. We also prove that Kr(R, b) = Kr(D, $*_c$) if and only if D is a $P{\ast}MD$. As a corollary, we have that if D is not a $P{\ast}MD$, then Kr(R, b) is an example of a kfr with respect to K(Y) and X but not a Kronecker function ring with respect to K(Y) and X.

• Kyungpook Mathematical Journal
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• 제56권4호
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• pp.1103-1113
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• 2016

In this paper, we study a directed simple graph ${\Gamma}_S(N)$ for a near-ring N, where the set $V^*(N)$ of vertices is the set of all left N-subsets of N with nonzero left annihilators and for any two distinct vertices $I,J{\in}V^*(N)$, I is adjacent to J if and only if IJ = 0. Here, we deal with the diameter, girth and coloring of the graph ${\Gamma}_S(N)$. Moreover, we prove a sufficient condition for occurrence of a regular element of the near-ring N in the left annihilator of some vertex in the strong zero-divisor graph ${\Gamma}_S(N)$.

• 대한수학회보
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• 제39권2호
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• pp.317-326
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• 2002

In this paper, we establish necessary and sufficient conditions for an exchange ring R to satisfy the n-stable range condition. It is shown that an exchange ring R satisfies the n-stable range condition if and only if for any regular a $\in$ R$^n$, there exists a unimodular u $\in$$^n$ R such that au $\in$ R is a group member, and if and only if whenever a$\simeq$$_n$b with a $\in$ R, b $\in$ M$_n$(R), there exist u $\in$ R$^n$, v $\in$$^n$ R such that a = ubv with uv = 1. As an application, we observe that exchange rings satisfying the n-stable range condition can be characterized by Drazin inverses. These also give nontrivial generalizations of [7, Theorem 10], [13, Theorem 10], [15, Theorem] and [16, Theorem. 2A].

• 호남수학학술지
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• 제31권4호
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• pp.515-527
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• 2009

Let R be a commutative ring with identity. Then we prove $M_n(R)=GL_n(R)$ ${\cup}${$A{\in}M_n(R)\;{\mid}\;detA{\neq}0$ and det $A{\neq}U(R)$}${\cup}Z(M-n(R))$ where U(R) denotes the set of all units of R. In particular, it will be proved that the full matrix ring $M_n(F)$ over a field F is the disjoint union of the general linear group $GL_n(F)$ of degree n over the field F and the set $Z(M_n(F))$ of all zero-divisors of $M_n(F)$. Using the result and universal mapping property we prove that $M_n(F)$ is its total ring of fractions.