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

• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.381-401
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• 2016

According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and ${\sigma}:M{\rightarrow}End(R)$ a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and ${\sigma}:M{\rightarrow}End(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).

• Korean Journal of Mathematics
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• v.22 no.2
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• pp.279-288
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• 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
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• v.59 no.4
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• pp.853-868
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• 2022

We study the structure of right regular commutators, and call a ring R strongly C-regular if ab - ba ∈ (ab - ba)²R for any a, b ∈ R. We first prove that a noncommutative strongly C-regular domain is a division algebra generated by all commutators; and that a ring (possibly without identity) is strongly C-regular if and only if it is Abelian C-regular (from which we infer that strong C-regularity is left-right symmetric). It is proved that for a strongly C-regular ring R, (i) if R/W(R) is commutative, then R is commutative; and (ii) every prime factor ring of R is either a commutative domain or a noncommutative division ring, where W(R) is the Wedderburn radical of R.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1603-1608
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• 2011

Let R be a ring with identity. If a, b, $c{\in}R$ such that a+b+c = 1, then the distributive laws from addition over multiplication hold in R, that is a+(bc) = (a+b)(a+c) when ab = ba, and (ab)+c = (a+c)(b+c) when ac = ca. An application to obtains, if A,B are idempotent matrices and AB = BA = 0 then there exists an idempotent matrix C such that A + BC = (A + B)(A + C), and also A + BC = (I - C)(I - B). Some other cases and applications are also presented.

• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.217-232
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• 2016

We make a study of two generalizations of regular rings, concentrating our attention on the structure of idempotents. A ring R is said to be right attaching-idempotent if for $a{\in}R$ there exists $0{\neq}b{\in}R$ such that ab is an idempotent. Next R is said to be generalized regular if for $0{\neq}a{\in}R$ there exist nonzero $b{\in}R$ such that ab is a nonzero idempotent. It is first checked that generalized regular is left-right symmetric but right attaching-idempotent is not. The generalized regularity is shown to be a Morita invariant property. More structural properties of these two concepts are also investigated.

• Bulletin of the Korean Chemical Society
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• v.18 no.10
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• pp.1076-1082
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• 1997

The infrared, Raman, and jet-cooled laser-induced fluorescence excitation spectra of 1,2,3,4-tetrahydronaphthalene have been recorded and analyzed. The observed vibrations have been assigned to understand the conformational behaviors in its electronic ground (S0) and excited (S1) states. Ab initio at the HF/6-31G** level and molecular mechanics (MM3) force field calculations have been carried out to generate the complete normal mode frequencies of the molecule in its S0 state. The vibrational frequencies calculated from the ab initio method show a better agreement with the observed infrared and Raman frequencies than those calculated from the MM3 method. In several cases, the normal mode calculations were very helpful to clarify some ambiguities of previous assignments. In addition, the ring inversion process between two twisted conformers of 1,2,3,4-tetrahydronaphthalene has been reexamined utilizing ab initio calculation. The results show that the ring inversion energy is in the range of 3.7-4.3 kcal/mol which is higher than the previously reported AM1 value of 2.1 kcal/mol.

• Bulletin of the Korean Chemical Society
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• v.15 no.8
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• pp.658-662
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• 1994

Ab initio and extended Huckel calculations have been applied to discuss the electronic structure, ring inversion barrier, and geometry of the $Cp_2TiS_3$ compound. The deformation of four membered ring in the planar geometry is originated from a second-order Jahn-Teller distortion due to the small energy gap between HOMO and LUMO on the basis of extended Huckel calculations. The puckered $C_s$ geometry is stabilized by the interaction of the $x^2-y^2$ metal orbital with the hybrid orbital in sulfur. Ab initio calculations have been carried out to explore the ring inversion process for the model $Cl_2TiS_3$ compound. We have optimized $C_s$ and $C_{2v}$ structures of the model compound at the RHF level. The energy barriers for the ring inversion are sensitive to the used basis set. With 4-31$G^*$ for the Cl and S ligands, the barriers are computed to be 8.41 kcal/mol at MP2 and 8.02 kcal/mol at MP4 level.

• Bulletin of the Korean Chemical Society
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• v.23 no.8
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• pp.1053-1054
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• 2002

• East Asian mathematical journal
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• v.30 no.5
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• pp.617-623
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• 2014

Due to Bell, a ring R is usually said to be IFP if ab = 0 implies aRb = 0 for $a,b{\in}R$. It is shown that if f(x)g(x) = 0 for $f(x)=a_0+a_1x$ and $g(x)=b_0+{\cdots}+b_nx^n$ in R[x], then $(f(x)R[x])^{2n+2}g(x)=0$. Motivated by this results, we study the structure of the IFP when proper ideals are taken in place of R, introducing the concept of insertion-of-ideal-factors-property (simply, IIFP) as a generalization of the IFP. A ring R will be called an IIFP ring if ab = 0 (for $a,b{\in}R$) implies aIb = 0 for some proper nonzero ideal I of R, where R is assumed to be non-simple. We in this note study the basic structure of IIFP rings.