• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1071-1083
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• 2023

The main motivation of this paper is to introduce and study the notions of strong Dedekind rings and semi-regular injective modules. Specifically, a ring R is called strong Dedekind if every semi-regular ideal is Q₀-invertible, and an R-module E is called a semi-regular injective module provided Ext¹_R(T, E) = 0 for every 𝓠-torsion module T. In this paper, we first characterize rings over which all semi-regular injective modules are injective, and then study the semi-regular injective envelopes of R-modules. Moreover, we introduce and study the semi-regular global dimensions sr-gl.dim(R) of commutative rings R. Finally, we obtain that a ring R is a DQ-ring if and only if sr-gl.dim(R) = 0, and a ring R is a strong Dedekind ring if and only if sr-gl.dim(R) ≤ 1, if and only if any semi-regular ideal is projective. Besides, we show that the semi-regular dimensions of strong Dedekind rings are at most one.

• Bulletin of the Korean Chemical Society
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• v.35 no.10
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• pp.3001-3004
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• 2014

A series of hydroxyl-derivatized quaternized polymers were successfully synthesized by atom transfer radical polymerization (ATRP) and Cu(I)-catalyzed 1,3-dipolar cycloaddition of azide and alkynes (click chemistry), followed by quaternization reactions. ATRP was employed to synthesize poly(2-hydroxyethyl methacrylate) (PHEMA), followed by introduction of alkyne groups using pentynoic acid, leading to HEMA-Alkyne. 2-Azido-1-ethanol and 3-azido-1-propanol were combined with the HEMA-Alkyne backbone via click reaction, resulting in triazole-ring containing hydroxyl-derivatized polymers. Quaternization reactions with methyl iodide were conducted on the triazole ring of each polymer. Molecular weight, molecular weight distribution, and the degree of quaternization (DQ) were determined by gel permeation chromatography (GPC) and $^1H$ NMR spectroscopy. The average molecular weight ($M_n$) of the resulting polymers ranged from $5.9{\times}10^4$ to $1.05{\times}10^5g/mol$ depending on the molecular architecture. The molecular weight distribution was low ($M_w/M_n$ = 1.26-1.38). The transmission spectra of the 0.1 wt % aqueous solutions of the resulting quaternized polymers at 650 nm were measured as a function of temperature. Results showed that the upper critical solution temperature (UCST) could be finely controlled by the level of DQ.