• Korean Journal of Mathematics
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• v.20 no.3
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• pp.273-284
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• 2012

A ring R is called reversibly Armendariz if $b_ja_i=0$ for all $i$, $j$ whenever $f(x)g(x)=0$ for two polynomials $f(x)=\sum_{i=0}^{m}a_ix^i,\;g(x)=\sum_{j=0}^{n}b_jx^j$ over R. It is proved that a ring R is reversibly Armendariz if and only if its polynomial ring is reversibly Armendariz if and only if its Laurent polynomial ring is reversibly Armendariz. Relations between reversibly Armendariz rings and related ring properties are examined in this note, observing the structures of many examples concerned. Various kinds of reversibly Armendariz rings are provided in the process. Especially it is shown to be possible to construct reversibly Armendariz rings from given any Armendariz rings.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.289-309
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• 2019

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.

• 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 $\alpha$.

• Communications of the Korean Mathematical Society
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• v.35 no.2
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• pp.447-454
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• 2020

The studies of reversible and NI rings have done important roles in noncommutative ring theory. A ring R shall be called QRUR if ab = 0 for a, b ∈ R implies that ba is contained in the upper nilradical of R, which is a generalization of the NI ring property. In this article we investigate the structure of QRUR rings and examine the QRUR property of several kinds of ring extensions including matrix rings and polynomial rings. We also show that if there exists a weakly semicommutative ring but not QRUR, then Köthe's conjecture does not hold.

• Bulletin of the Korean Chemical Society
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• v.29 no.4
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• pp.827-832
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• 2008

A new diarylethene compound with donor and acceptor substituent was synthesized from 2,3-bis(2-methylbenzo[b]thiophene-3-yl)hexafluorocyclopentene (BTF) over 5 steps. The donor-acceptor structured BTF compound (TBTFE) showed spectral change to a longer wavelength through photochromism with a high cyclization quantum yield (0.56). The 3,4-ethylenedioxythiophene (T) and carboethoxy (E) groups directly connected to BTF unit promoted electrical change accompanied with the photoisomerization of the BTF unit. Photo-induced electrical switching was achieved from a photocell containing TBTFE doped polymer film, which showed reversible and stable current change over repeated cycles by the alternative UV/Vis irradiation, as estimated by the I-V plot.

• Journal of the Korean Mathematical Society
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• v.51 no.4
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• pp.751-771
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• 2014

We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring properties pass to power series rings and polynomial rings. It is also shown that ${\pi}$-regular rings are strongly ${\pi}$-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to construct such kinds of noncommutative rings appropriate for the situations occurred naturally in studying standard ring theoretic properties.

• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.415-427
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• 2019

Let R be an associative ring with nonzero identity. The annihilator ideal graph of R, denoted by ${\Gamma}_{Ann}(R)$, is a graph whose vertices are all nonzero proper left ideals and all nonzero proper right ideals of R, and two distinct vertices I and J are adjacent if $I{\cap}({\ell}_R(J){\cup}r_R(J)){\neq}0$ or $J{\cap}({\ell}_R(I){\cup}r_R(I)){\neq}0$, where ${\ell}_R(K)=\{b{\in}R|bK=0\}$ is the left annihilator of a nonempty subset $K{\subseteq}R$, and $r_R(K)=\{b{\in}R|Kb=0\}$ is the right annihilator of a nonempty subset $K{\subseteq}R$. In this paper, we assume that R is a semicommutative ring. We study the structure of ${\Gamma}_{Ann}(R)$. Also, we investigate the relations between the ring-theoretic properties of R and graph-theoretic properties of ${\Gamma}_{Ann}(R)$. Moreover, some combinatorial properties of ${\Gamma}_{Ann}(R)$, such as domination number and clique number, are studied.

• Bulletin of the Korean Chemical Society
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• v.22 no.12
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• pp.1316-1322
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• 2001

Transition metal complexes of novel mono- and di-ferrocene-substituted porphyrins have been synthesized and characterized by structural and electrochemical methods. The X-ray structures of Mn(FPTTP)Cl and Mn(DFTTP)Cl showed the distorted square pyramidal coordination geometry with syn configuration of chloride and ferrocenyl substituents. The electrochemistry of ferrocene-substituted porphyrins and their metal complexes has been determined to elucidate the ${\pi}-conjugation$ effect of the porphyrin ring. The ferrocenyl group of H2FPTTP underwent a reversible one-electron transfer process at 0.30 V, indicating the good electron donating effect of the phorphyrin ring to the ferrocene substituent. The redox potential of the ferrocenyl subunit and porphyrin ring was affected by the central metal ions of the metalloporphyrins, that is, Zn(II) and Ni(II) made the oxidation of ferrocene much easier and Mn(III) made it harder. The ferrocene subunits of H2DFTTP interacted electrochemically with each other with peak splitting of 0.21 V. The strength of the electrochemical interactions between the two ferrocenyl substituents can be controlled by central metal ions of metalloporphyrins.

• Bulletin of the Korean Chemical Society
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• v.28 no.11
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• pp.1996-2002
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• 2007

The redox behavior of a [Ni6(μ3-Se)2(μ4-Se)3(Fe(η 5-C5H4P-Ph2)2)3] (= [Ni-Se-dppf], dppf = 1,1-bis(diphenylphosphino) ferrocene) cluster was studied using platinum (Pt) and glassy carbon electrodes (GCE) in nonaqueous media. The cluster showed electrochemical activity at the potential range between +1.6 and ?1.6 V. In the negative region (0 to ?1.6 V), the cluster exhibited two-step reductions. The first step was one-electron reversible, while the second step was a five-electron quasi-reversible process. On the other hand, in the positive region (0 to +1.6 V), the first step involved one-electron quasi-reversible process. The applicability of the cluster was found towards the electrocatalytic reduction of CO2 and was evaluated by experiments using rotating ring disc electrode (RRDE). RRDE experiments demonstrated that two electrons were involved in the electrocatalytic reduction of CO2 to CO at the Se-Ni-dppf-modified electrode.

• Journal of applied mathematics & informatics
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• v.33 no.3_4
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• pp.469-474
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• 2015

In this paper, we will consider the notions of (m, n)-potent conditions in near-rings, in particular, a near-ring R with left bipotent or right bipotent condition. We will derive some properties of near-rings with (1, n) and (n, 1)-potent conditions where n is a positive integer, and then some properties of near-rings with (m, n)-potent conditions. Also, we may discuss the behavior of R-subgroups in (1, n)-potent or (n, 1)-potent near-rings..