• Bulletin of the Korean Mathematical Society
• /
• v.56 no.5
• /
• pp.1257-1272
• /
• 2019

This article concerns the structure of idempotents in reversible and pseudo-reversible rings in relation with various sorts of ring extensions. It is known that a ring R is reversible if and only if $ab{\in}I(R)$ for $a,b{\in}R$ implies ab = ba; and a ring R shall be said to be pseudoreversible if $0{\neq}ab{\in}I(R)$ for $a,b{\in}R$ implies ab = ba, where I(R) is the set of all idempotents in R. Pseudo-reversible is seated between reversible and quasi-reversible. It is proved that the reversibility, pseudoreversibility, and quasi-reversibility are equivalent in Dorroh extensions and direct products. Dorroh extensions are also used to construct several sorts of rings which are necessary in the process.

• Bulletin of the Korean Mathematical Society
• /
• v.56 no.4
• /
• pp.993-1006
• /
• 2019

This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring R shall be said to be quasi-reversible if $0{\neq}ab{\in}I(R)$ for a, $b{\in}R$ implies $ba{\in}I(R)$, where I(R) is the set of all idempotents in R. We investigate the quasi-reversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility.

• Kyungpook Mathematical Journal
• /
• v.60 no.1
• /
• pp.71-72
• /
• 2020

A ring R is quasi-reversible if 0 ≠ ab ∈ I(R) for a, b ∈ R implies ba ∈ I(R), where I(R) is the set of all idempotents in R. In this short paper, we prove that the ring of 2×2 matrices over an arbitrary field is quasi-reversible, which is an answer to the question given by Da Woon Jung et al. in [Bull. Korean Math. Soc., 56(4) (2019) 993-1006].

• Korean Journal of Mathematics
• /
• v.22 no.2
• /
• pp.279-288
• /
• 2014

The studies of reversible and 2-primal rings have done important roles in noncommutative ring theory. We in this note introduce the concept of quasi-reversible-over-prime-radical (simply, QRPR) as a generalization of the 2-primal ring property. A ring is called QRPR if ab = 0 for $a,b{\in}R$ implies that ab is contained in the prime radical. In this note we study the structure of QRPR rings and examine the QRPR property of several kinds of ring extensions which have roles in noncommutative ring theory.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.6
• /
• pp.1913-1925
• /
• 2017

Let R be a ring with identity. An ideal N of R is called ideal-symmetric (resp., ideal-reversible) if $ABC{\subseteq}N$ implies $ACB{\subseteq}N$ (resp., $AB{\subseteq}N$ implies $BA{\subseteq}N$) for any ideals A, B, C in R. A ring R is called ideal-symmetric if zero ideal of R is ideal-symmetric. Let S(R) (called the ideal-symmetric radical of R) be the intersection of all ideal-symmetric ideals of R. In this paper, the following are investigated: (1) Some equivalent conditions on an ideal-symmetric ideal of a ring are obtained; (2) Ideal-symmetric property is Morita invariant; (3) For any ring R, we have $S(M_n(R))=M_n(S(R))$ where $M_n(R)$ is the ring of all n by n matrices over R; (4) For a quasi-Baer ring R, R is semiprime if and only if R is ideal-symmetric if and only if R is ideal-reversible.

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

• Bulletin of the Korean Chemical Society
• /
• v.17 no.2
• /
• pp.173-179
• /
• 1996

A series of tetradentate Schiff base ligands; [1,2-bis(naphthylideneimino)ethane, 1,3-bis(naphthylideneimino)propane, 1,4-bis(naphthylideneimino)butane, and 1,5-bis(naphthylideneimino)pentane] and their Ni(Ⅱ) complexes have been synthesized. The properties of these ligands and their Ni(Ⅱ) complexes have been characterized by elemental analysis, IR, NMR, UV-vis spectra, molar conductance, and thermogravimetric analysis. The mole ratio of Schiff base to Ni(Ⅱ) metal was found to be 1:1. The electrochemical redox process of the ligands and their Ni(Ⅱ) complexes in DMF and DMSO solution containing 0.1 M tetraethyl ammonium perchlorate (TEAP) as a supporting electrolyte have been investigated by cyclic voltammetry, chronoamperometry, differential pulse voltammetry, and controlled potential coulometry at glassy carbon electrode. The redox process of the ligands was highly irreversible, whereas redox process of Ni(Ⅱ) complexes were observed as one electron transfer process in quasi-reversible and diffusion-controlled reaction. The electrochemical redox potentials of the Ni(Ⅱ) complexes were affected by the chelate ring size of ligands. The diffusion coefficients of Ni(Ⅱ) complexes containing 0.1 M TEAP in DMSO solution were determined to be 5.7-6.9 × 10-6 cm2/sec. Also the exchange rate constants were determined to be 1.8-9.5 × 10-2 cm2/sec. These values were affected by the chelate ring size of ligands.

• Bulletin of the Korean Chemical Society
• /
• v.17 no.8
• /
• pp.688-693
• /
• 1996

Tetradentate Schiff base ligands derived from 2-hydroxy-1-naphthaldehyde and aliphatic diamine have been synthesized. Cu(Ⅱ) complexes of Schiff base ligands have been synthesized from the free ligands and copper acetate. The mole ratio of ligand to copper was identified to be 1:1 by the result of elemental analysis and Cu(Ⅱ) complexes were in a four-coordinated configuration. The electrochemical redox process of Cu(Ⅱ) complexes in a DMF solution has been investigated by cyclic voltammetry, chronoamperometry, differential pulse voltammetry, and controlled potential coulometry. The redox process of Cu(Ⅱ) complexes is one electron transfer process in quasi-reversible and diffusion-controlled reaction. The electrochemical redox potentials and the kinetic parameters of Cu(Ⅱ) complexes are affected by the chelate ring of Schiff base ligands.

• Bulletin of the Korean Chemical Society
• /
• v.18 no.8
• /
• pp.850-856
• /
• 1997

A series of tetradentate Schiff-base ligands; 1,3-bis(salicylideneimino) propane, 1,4-bis(salicylideneimino)butane, and 1,5-bis(salicylideneimino)pentane, and their Cu(Ⅱ) and Ni(Ⅱ) complexes have been synthesized. The properties of ligands and complexes have been characterized by elemental analysis, IR, NMR, UV-Vis spectra, molar conductance, and thermogravimetric anaylsis. The mole ratio of Schiff base to metal at complexes was found to be 1 : 1. All complexes were four-coordinated configuration and non-ionic compound. The electrochemical redox processes of the ligands and their complexes in DMF solution containing 0.1 M TEAP as supporting electrolyte have been investigated by cyclic voltammetry, chronoamperometry, differential pulse voltammetry at glassy carbon electrode, and by controlled potential coulometry at platinum gauze electrode. The redox process of the ligands was highly irreversible, whereas redox process of Cu(Ⅱ) and Ni(Ⅱ) complexes was observed as one electron transfer process of quasi-reversible and diffusion-controlled reaction. Also the electrochemical redox potentials of complexes were affected by chelate ring size of ligands. The diffusion coefficients of Cu(Ⅱ) and Ni(Ⅱ) complexes in DMF solution were determined to be 4.2-6.6×10-6 cm2/sec. Also the exchange rate constants were determined to be 3.6-9.7×10-2 cm/sec.

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