• East Asian mathematical journal
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• v.34 no.5
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• pp.615-621
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• 2018

Let R be a ring with the unity 1, and let e be an idempotent of R. In this paper, we discuss some symmetric property for the set $\{(a_1,a_2,{\cdots},a_n){\in}R^n:a_1a_2{\cdots}a_n=e\}$. We here investigate some properties of those rings with such a symmetric property for an arbitrary idempotent e; some of our results turn out to generalize some known results observed already when n = 2 and e = 0, 1 by several authors. We also focus especially on the case when n = 3 and e = 1. As consequences of our observation, we also give some equivalent conditions to the commutativity for some classes of rings, in terms of the symmetric property.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.57-71
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• 2019

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and ${\omega}:M{\rightarrow}End(R)$ a monoid homomorphism. Let R be a reversible, M-compatible ring and ${\alpha}=a_1g_1+{\cdots}+a_ng_n$ a non-zero element in skew monoid ring $R{\ast}M$. It is proved that if there exists a non-zero element ${\beta}=b_1h_1+{\cdots}+b_mh_m$ in $R{\ast}M$ with ${\alpha}{\beta}=c$ is a constant, then there exist $1{\leq}i_0{\leq}n$, $1{\leq}j_0{\leq}m$ such that $g_{i_0}=e=h_{j_0}$ and $a_{i_0}b_{j_0}=c$ and there exist elements a, $0{\neq}r$ in R with ${\alpha}r=ca$. As a consequence, it is proved that ${\alpha}{\in}R*M$ is unit if and only if there exists $1{\leq}i_0{\leq}n$ such that $g_{i_0}=e$, $a_{i_0}$ is unit and aj is nilpotent for each $j{\neq}i_0$, where R is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of R and those elements in skew monoid ring $R{\ast}M$, where R is a reversible or right duo ring.

• 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.

• Bulletin of the Korean Chemical Society
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• v.15 no.11
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• pp.980-984
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• 1994

Monomeric rhodium and iridium diimine complexes $Cp^*M(HNRNH)(Cp^*$ = 1,2,3,4,5-pentamethylcyclopentadienyl : (M=lr; R=o-$C_6H_4 (1a), 4,5-(CH_3)_2-C_6H_2-1,2 (1b), 4,5-(Cl)_2-C_6H_2-1,2$ (1c), NCC=CCN-1,2 (1d): M=Rh; R=NCC=CCN-1,2 (1e)) have been synthesized from $[CP^*MCl_2]_2$ and 2 equiv. of diamine in the presence of $NEt_3$. The Crystal structure of 1a was determined by X-ray diffraction method : 1a was crystallized in the monoclinic system, space group $P2_{1/c}$, with lattice constants a=9.543 (1) ${\AA}$, b=16.286 (1) ${\AA}$, c=10.068 (1) ${\AA}$ and ${\beta}$=99.25 (1), with Z= 4. Least-squares refinement of the structure led to R factor of 0.049. The coordination sphere of rhodium and iridium can be described as a 2-legged piano-stool. All complexes are highly colored. Electrochemical studies show that 1d and 1e display quasi-reversible reduction and 1a-1c display irreversible reductions, suggesting that the acceptor orbital might be localized on the diimine ring.