• Journal of the Korean Mathematical Society
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• v.45 no.3
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• pp.727-740
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• 2008

A ring R is called IFP, due to Bell, if ab=0 implies aRb=0 for $a,b{\in}R$. Huh et al. showed that the IFP condition need not be preserved by polynomial ring extensions. But it is shown that ${\sum}^n_{i=0}$ $E_{ai}E$ is a nonzero nilpotent ideal of E whenever R is an IFP ring and $0{\neq}f{\in}F$ is nilpotent, where E is a polynomial ring over R, F is a polynomial ring over E, and $a_i^{'s}$ are the coefficients of f. we shall use the term near IFP to denote such a ring as having place near at the IFPness. In the present note the structures of IFP rings and near-IFP rings are observed, extending the classes of them. IFP rings are NI (i.e., nilpotent elements form an ideal). It is shown that the near-IFPness and the NIness are distinct each other, and the relations among them and related conditions are examined.

• Journal of the Korean Mathematical Society
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• v.60 no.6
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• pp.1233-1254
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• 2023

We introduce two subclasses of abelian McCoy rings, so-called π-CN-rings and π-duo rings, and systematically study their fundamental characteristic properties accomplished with relationships among certain classical sorts of rings such as 2-primal rings, bounded rings etc. It is shown that a ring R is π-CN whenever every nilpotent element of index 2 in R is central. These rings naturally generalize the long-known class of CN-rings, introduced by Drazin [9]. It is proved that π-CN-rings are abelian, McCoy and 2-primal. We also show that, π-duo rings are strongly McCoy and abelian and also they are strongly right AB. If R is π-duo, then R[x] has property (A). If R is π-duo and it is either right weakly continuous or every prime ideal of R is maximal, then R has property (A). A π-duo ring R is left perfect if and only if R contains no infinite set of orthogonal idempotents and every left R-module has a maximal submodule. Our achieved results substantially improve many existing results.

• East Asian mathematical journal
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• v.36 no.3
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• pp.337-347
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• 2020

Let R be a ring with unity, and let I be an ideal of R. Then R is called I-semiregular if for every a ∈ R there exists b ∈ R such that ab is an idempotent of R and a - aba ∈ I. In this paper, basic properties of I-semiregularity are investigated, and some equivalent conditions to the primitivity of e are observed for an idempotent e of an I-semiregular ring R such that I∩eR = (0). For an abelian regular ring R with the ascending chain condition on annihilators of idempotents of R, it is shown that R is isomorphic to a direct product of a finite number of division rings, as a consequence of the observations.

• Proceeding of EDISON Challenge
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• 2013.04a
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• pp.205-209
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• 2013

본 실험에서는 탄소 cluster 중에서 fullerene 구조를 가질 수 있는 가장 작은 cluster인 $C_{20}$ cluster에 대해, 기존 연구에서 가장 안정한 것으로 제시된 cage(fullerene), bowl, ring의 3가지 구조와 $Si_{20}$ cluster를 모방한 구조 하나의 안정성을 확인하였다. ab-initio calculation을 지원하는 Edison nanophysics의 LCAODFLab을 이용하여 LDA-CA, GGA-PBE 두 가지 방법으로 계산하였다. 계산 값을 바탕으로 각 구조의 원자화에너지를 비교한 결과 LDA와 GGA 모두 cage, bowl, ring의 순서로 안정하였다. 최적화한 구조에 대하여 구조분석을 진행하였다. 최적화 결과 Bowl은 $C_{5v}$, ring은 $D_{10h}$, cage는 $C_{2h}$ 대칭성을 가지는 구조였으며, LDA, GGA 계산 모두 $C_{20}$ 구조의 spin polarization에는 영향을 받지 않았다.

• Bulletin of the Korean Chemical Society
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• v.21 no.3
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• pp.300-304
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• 2000

The B3LYP method based on the density functional theory(DFT) is shown to be much better than the ab initio MP2 method for structural determination of diphosphadithiatetrazocine systems having transannular S--S bonding. The presence of bonding between the two sulfur atoms across the cyclic ring is theoretically confirmed in the case of the neutral diphosphadithiatetrazocine. The S--S dobding disappears in the ionized species. The planarity of the dicationic heterocyclic ring system turns out to be closely associated with the $\pi-electron$ delocalization over the entire ring as well as the N-S-N bonds, which become stiffened upon ionizaiton. In the case of dianionic species, the chair-boat and chair conformers are nearly degenerate and far more stable than the crown conformer.

• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.455-466
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• 2010

For an endomorphism $\alpha$ of a ring R, we introduce the weak $\alpha$-skew Armendariz rings which are a generalization of the $\alpha$-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak $\alpha$-skew Armendariz if and only if for any n, the $n\;{\times}\;n$ upper triangular matrix ring $T_n(R)$ is weak $\bar{\alpha}$-skew Armendariz, where $\bar{\alpha}\;:\;T_n(R)\;{\rightarrow}\;T_n(R)$ is an extension of $\alpha$ If R is reversible and $\alpha$ satisfies the condition that ab = 0 implies $a{\alpha}(b)=0$ for any a, b $\in$ R, then the ring R[x]/($x^n$) is weak $\bar{\alpha}$-skew Armendariz, where ($x^n$) is an ideal generated by $x^n$, n is a positive integer and $\bar{\alpha}\;:\;R[x]/(x^n)\;{\rightarrow}\;R[x]/(x^n)$ is an extension of $\alpha$. If $\alpha$ also satisfies the condition that ${\alpha}^t\;=\;1$ for some positive integer t, the ring R[x] (resp, R[x; $\alpha$) is weak $\bar{\alpha}$-skew (resp, weak) Armendariz, where $\bar{\alpha}\;:\;R[x]\;{\rightarrow}\;R[x]$ is an extension of $\alpha$.

• Journal of the Korean Chemical Society
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• v.41 no.3
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• pp.123-129
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• 1997

In order to consider the reduced mass effects on the out-of-plane ring inversion vibration of 1,3-CHD, the vector-based computer program has been written and the kinetic energy expansion function for the large amplitude ring inversion vibration has been calculated using this program. The structural parameters for the calculations have been determined from the ab initio HF/6-31G** calculation. The potential energy function for the out-of-plane ring inversion vibration of 1,3-CHD has been determined from the kinetic energy expansion function and previously reported low-frequency Raman data. The vibrational Hamiltonian calculation including kinetic energy expansion function made it possible to determine the more reliable out-of-plane potential energy function for the ring inversion of 1,3-CHD.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.437-453
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• 2022

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ~ on S defined by letting a ~ b if and only if ann_S(a) = ann_S(b), is an equivalence relation. The compressed zero-divisor graph 𝚪_E(S) of S is the undirected graph whose vertices are the equivalence classes induced by ~ on S other than [0]_S and [1]_S, in which two distinct vertices [a]_S and [b]_S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R₀[x] and the near-ring of the power series R₀[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(𝚪_E(R₀[x])) and diam(𝚪_E(R₀[[x]])). As a corollary, it is shown that for a reduced ring R, diam(𝚪_E(R)) ≤ diam(𝚪_E(R₀[x])) ≤ diam(𝚪_E(R₀[[x]])).

• Communications of the Korean Mathematical Society
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• v.27 no.1
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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 $\Gamma(Z_nQ_8)$.

• Bulletin of the Korean Mathematical Society
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• v.57 no.4
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• pp.957-972
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• 2020

Let 𝓡 be a 2-torsion free unital ring containing a non-trivial idempotent. An additive map 𝛿 from 𝓡 into itself is called a Jordan derivable map at commutative zero point if 𝛿(AB + BA) = 𝛿(A)B + B𝛿(A) + A𝛿(B) + 𝛿(B)A for all A, B ∈ 𝓡 with AB = BA = 0. In this paper, we prove that, under some mild conditions, each Jordan derivable map at commutative zero point has the form 𝛿(A) = 𝜓(A) + CA for all A ∈ 𝓡, where 𝜓 is an additive Jordan derivation of 𝓡 and C is a central element of 𝓡. Then we generalize the result to the case of Jordan higher derivable maps at commutative zero point. These results are also applied to some operator algebras.