• Communications of the Korean Mathematical Society
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• v.38 no.2
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• pp.365-376
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• 2023

The goal of this article is to present the graded J-ideals of G-graded rings which are extensions of J-ideals of commutative rings. A graded ideal P of a G-graded ring R is a graded J-ideal if whenever x, y ∈ h(R), if xy ∈ P and x ∉ J(R), then y ∈ P, where h(R) and J(R) denote the set of all homogeneous elements and the Jacobson radical of R, respectively. Several characterizations and properties with supporting examples of the concept of graded J-ideals of graded rings are investigated.

• 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 Mathematical Society
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• v.60 no.4
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• pp.895-903
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• 2023

In this paper, we give some results on 2-strongly Gorenstein projective modules and related rings. We first investigate the relationship between strongly Gorenstein projective modules and periodic modules and then give the structure of modules over strongly Gorenstein semisimple rings. Furthermore, we prove that a ring R is 2-strongly Gorenstein hereditary if and only if every ideal of R is Gorenstein projective and the class of 2-strongly Gorenstein projective modules is closed under extensions. Finally, we study the relationship between 2-Gorenstein projective hereditary and 2-Gorenstein projective semisimple rings, and we also give an example to show the quotient ring of a 2-Gorenstein projective hereditary ring is not necessarily 2-Gorenstein projective semisimple.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.93-104
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• 2024

Let R be a commutative ring with identity. In this paper, we introduce a new class of ideals called the class of strongly quasi J-ideals lying properly between the class of J-ideals and the class of quasi J-ideals. A proper ideal I of R is called a strongly quasi J-ideal if, whenever a, b ∈ R and ab ∈ I, then a² ∈ I or b ∈ Jac(R). Firstly, we investigate some basic properties of strongly quasi J-ideals. Hence, we give the necessary and sufficient conditions for a ring R to contain a strongly quasi J-ideals. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the primary ideals, the prime ideals and the maximal ideals. Finally, we give an idea about some strongly quasi J-ideals of the quotient rings, the localization of rings, the polynomial rings and the trivial rings extensions.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.45-58
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• 2024

Let R be a commutative ring with identity. If the nilpotent radical N il(R) of R is a divided prime ideal, then R is called a ϕ-ring. Let R be a ϕ-ring and S be a multiplicative subset of R. In this paper, we introduce and study the class of nonnil-S-coherent rings, i.e., the rings in which all finitely generated nonnil ideals are S-finitely presented. Also, we define the concept of ϕ-S-coherent rings. Among other results, we investigate the S-version of Chase's result and Chase Theorem characterization of nonnil-coherent rings. We next study the possible transfer of the nonnil-S-coherent ring property in the amalgamated algebra along an ideal and the trivial ring extension.

• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.973-989
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• 2013

We observe from known results that the set of nilpotent elements in Armendariz rings has an important role. The upper nilradical coincides with the prime radical in Armendariz rings. So it can be shown that the factor ring of an Armendariz ring over its prime radical is also Armendariz, with the help of Antoine's results for nil-Armendariz rings. We study the structure of rings with such property in Armendariz rings and introduce APR as a generalization. It is shown that APR is placed between Armendariz and nil-Armendariz. It is shown that an APR ring which is not Armendariz, can always be constructed from any Armendariz ring. It is also proved that a ring R is APR if and only if so is R[$x$], and that N(R[$x$]) = N(R)[$x$] when R is APR, where R[$x$] is the polynomial ring with an indeterminate $x$ over R and N(-) denotes the set of all nilpotent elements. Several kinds of APR rings are found or constructed in the precess related to ordinary ring constructions.

• Structural Engineering and Mechanics
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• v.49 no.5
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• pp.619-629
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• 2014

As hyperboloidal cooling towers (HCTs) growing larger and slender, they become more sensitive to gust wind. To improve the dynamic properties of HCTs and to improve the wind resistance capability, stiffening rings have been studied and applied. Although there have been some findings, the influence mechanism of stiffening rings on the dynamic properties is still not fully understood. Based on some fundamental perceptions on the dynamic properties of HCTs and free ring structures, a concept named "participation degree" of stiffening rings was proposed and the influence mechanism on the dynamic properties was illustrated. The "participation degree" is determined by the modal deform amplitude and latitude wave number of stiffening rings. Larger modal deform amplitude and more latitude waves can both result in higher participation degree and more improvement to eigenfrequencies. Also, this concept can explain and associate the pre-existing independent findings.

• Journal of the Korean Institute of Gas
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• v.16 no.5
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• pp.52-57
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• 2012

This study presents sealing characteristics of multi-contact o-rings as functions of strain, compression stress, and contact normal stress using a FEM technique. The FEM results on the sealing characteristics show that the maximum strain, maximum compression stress, and maximum contact normal stress of multi-contact o-rings are approximately 1.7 times higher than those of conventional o-rings. This is due to a U-grooved cross section of multi-contact o-rings, and the multi-contact o-rings with a U-groove show more effective in sealing for high pressure vessels, valves, and gas equipments. And the extrusion failure in the multi- contact o-ring does not produce for an increased gas pressure due to a U-groove. This may extend sealing life compared to that of a conventional o-ring.

• 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 Society of Manufacturing Process Engineers
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• v.19 no.3
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• pp.22-28
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• 2020

This paper investigates the effects of retaining ring materials, particularly PPS and PEEK, used in the CMP process, on wafer polishing and ring wear. CMP can be performed using bonded type retaining rings made with PPS or injection molding type retaining rings made with PEEK. In this study, after polishing a wafer with a PPS retaining ring, the average profile height of the wafer was 0.098 ㎛ less than that of the wafer polished with a PEEK retaining ring, implying that PPS retaining rings achieve a higher polishing rate. In addition, the center area of the wafer profile had less deviation and improved flatness after polishing with the PPS ring. These results indicate that a higher polishing rate and smaller profile height deviation can be achieved using retaining rings made with PPS compared to retaining rings made with PEEK. Therefore, with semiconductor circuit patterns becoming finer and wafer sizes becoming larger, the use of PPS in CMP retaining rings can obtain more stable wafer polishing results compared to that of PEEK.