• Bulletin of the Korean Mathematical Society
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• v.60 no.6
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• pp.1523-1537
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• 2023

Let R be a commutative ring with identity and S a multiplicative subset of R. First, we introduce and study the u-S-projective dimension and u-S-injective dimension of an R-module, and then explore the u-S-global dimension u-S-gl.dim(R) of a commutative ring R, i.e., the supremum of u-S-projective dimensions of all R-modules. Finally, we investigate u-S-global dimensions of factor rings and polynomial rings.

• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.97-112
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• 2023

In this paper, we introduce and study the u-S-weak global dimension u-S-w.gl.dim(R) of a commutative ring R for some multiplicative subset S of R. Moreover, the u-S-weak global dimensions of factor rings and polynomial rings are investigated.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.173-187
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• 2014

In this article, we introduce and study the Gorenstein cotorsion dimension of modules and rings. It is shown that this dimension has nice properties when the ring in question is left GF-closed. The relations between the Gorenstein cotorsion dimension and other homological dimensions are discussed. Finally, we give some new characterizations of weak Gorenstein global dimension of coherent rings in terms of Gorenstein cotorsion modules.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1327-1338
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• 2015

In this paper, we introduce and study the w-weak global dimension w-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a $Pr\ddot{u}fer$ v-multiplication domain if and only if w-w.gl.dim(R) ${\leq}1$. We also show that there is a large class of domains in which Hilbert's syzygy Theorem for the w-weak global dimension does not hold. Namely, we prove that if R is an integral domain (but not a field) for which the polynomial ring R[x] is w-coherent, then w-w.gl.dim(R[x]) = w-w.gl.dim(R).

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.735-744
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• 2024

In this note, we shed new light on Krull domains from the point view of Gorenstein homological algebra. By using the so-called w-operation, we show that an integral domain R is Krull if and only if for any nonzero proper w-ideal I, the Gorenstein global dimension of the w-factor ring (R/I)_w is zero. Further, we obtain that an integral domain R is Dedekind if and only if for any nonzero proper ideal I, the Gorenstein global dimension of the factor ring R/I is zero.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.59-69
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• 2024

Let R be a commutative ring. An R-module E is said to be regular injective provided that Ext¹_R(R/I, E) = 0 for any regular ideal I of R. We first show that the class of regular injective modules have the hereditary property, and then introduce and study the regular injective dimension of modules and regular global dimension of rings. Finally, we give some homological characterizations of total rings of quotients and Dedekind rings.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.961-976
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• 2019

Let R be any commutative ring and S be any multiplicative closed set. We introduce an S-version of $\mathcal{F}$-Mittag-Leffler modules, called $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and define the projective dimension with respect to these modules. We give some characterizations of $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, investigate the relationships between $\mathcal{F}$-Mittag-Leffler modules and $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler modules, and use these relations to describe noetherian rings and coherent rings, such as R is noetherian if and only if $R_S$ is noetherian and every $\mathcal{F}_{\mathcal{S}}$-Mittag-Leffler module is $\mathcal{F}$-Mittag-Leffler. Besides, we also investigate the $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension of R, and prove that $R_S$ is noetherian if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is zero; $R_S$ is coherent if and only if its $\mathcal{M}^{\mathcal{F}_{\mathcal{S}}$-global dimension is at most one.

• Communications of the Korean Mathematical Society
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• v.30 no.3
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• pp.201-208
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• 2015

This paper reviews recent mathematical progresses made on the study of the initial-value problem for nonlinear Dirac equations in one space dimension. We also prove the global existence of solutions to some nonlinear Dirac equations and propose a model problem (3.6).

• 제어로봇시스템학회:학술대회논문집
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• 1992.10a
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• pp.1102-1107
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• 1992

Existing CAD systems do nto provide the advanced function for systematic checking of design and drafting errors in mechanical drawings. This paper describes a method for sytematic checking of global parts in mechanical drawings. The checking items are deficiency and redundancy of dimensions, input-errors in dimension figures and symbols, etc. Checking for deficiency and redundancy of global dimensions has been performed applying Graph Theory. This system has been applied to several examples and we have confirmed the feasibility of this checing method.

• Journal of the Korean Society for Precision Engineering
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• v.13 no.8
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• pp.60-71
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• 1996

Mostof existing CAD systems do not provide the advanced function for systematic checking of design and drafting errors in mechanical drawings. We have reported a computer aided drawing check system to single plane projection drawings made by a CAD system. This paper describes a checking method of dimensioning errors in mechanical drawings. The checking items are deficiency and redundancy of dimensions, input-errors in dimension figures and symbols, etc. Checking for deficiency and redundancy of global dimensions has been performed applying Graph Theory. This system has been applied to several examples and we have confirmed the feasibility of this checking method.