• 대한수학회보
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• 제49권1호
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• pp.205-211
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• 2012

Let R be a graded Noetherian domain and A a graded Krull overring of R. We show that if h-dim $R\leq2$, then A is a graded Noetherian domain with h-dim $A\leq2$. This is a generalization of the well-know theorem that a Krull overring of a Noetherian domain with dimension $\leq2$ is also a Noetherian domain with dimension $\leq2$.

• 대한수학회보
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• 제51권2호
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• pp.519-530
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• 2014

Let (R,m) be a Noetherian local ring and M a finitely generated R-module. For an integer s > -1, we say that M is Cohen-Macaulay in dimension > s if every system of parameters of M is an M-sequence in dimension > s introduced by Brodmann-Nhan [1]. In this paper, we give some characterizations for Cohen-Macaulay modules in dimension > s in terms of the Noetherian dimension of the local cohomology modules $H^i_m(M)$, the polynomial type of M introduced by Cuong [5] and the multiplicity e($\underline{x}$;M) of M with respect to a system of parameters $\underline{x}$.

• 대한수학회보
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• 제56권4호
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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.

• 대한수학회지
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• 제59권4호
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• pp.821-841
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• 2022

In this paper, we introduce and study regular rings relative to the hereditary torsion theory w (a special case of a well-centered torsion theory over a commutative ring), called w-regular rings. We focus mainly on the w-regularity for w-coherent rings and w-Noetherian rings. In particular, it is shown that the w-coherent w-regular domains are exactly the Prüfer v-multiplication domains and that an integral domain is w-Noetherian and w-regular if and only if it is a Krull domain. We also prove the w-analogue of the global version of the Serre-Auslander-Buchsbaum Theorem. Among other things, we show that every w-Noetherian w-regular ring is the direct sum of a finite number of Krull domains. Finally, we obtain that the global weak w-projective dimension of a w-Noetherian ring is 0, 1, or ∞.

• 대한수학회논문집
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• 제38권4호
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• pp.983-992
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• 2023

Let A be a ring and 𝓙 = {ideals I of A | J(I) = I}. The Krull dimension of A, written dim A, is the sup of the lengths of chains of prime ideals of A; whereas the dimension of the maximal spectrum, denoted by dim _𝓙A, is the sup of the lengths of chains of prime ideals from 𝓙. Then dim _𝓙A ≤ dim A. In this paper, we will study the dimension of the maximal spectrum of some constructions of rings and we will be interested in the transfer of the property J-Noetherian to ring extensions.

• Korean Journal of Mathematics
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• 제7권2호
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• pp.167-169
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• 1999

Let R be a Noetherian domain. It is proved that $t$-dimR = 1 if and only if each (proper if R is not a valuation domain) $t$-linked overring D of R is of $t$-dimD = 1 if and only if each integrally closed $t$-linked overring of R is a Krull domain.

• 대한수학회지
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• 제58권5호
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• pp.1239-1260
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• 2021

We study the set of star operations on local Noetherian domains D of dimension 1 such that the conductor (D : T) (where T is the integral closure of D) is equal to the maximal ideal of D. We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension k ⊆ B, where k is a field, and then we study how the cardinality of this set of closures vary as the size of k varies while the structure of B remains fixed.

• 대한수학회논문집
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• 제33권4호
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• pp.1321-1329
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• 2018

In this paper, we compare the dimension of a constructible set with the dimension of its inverse image under morphisms of finite type of noetherian Jacobson schemes.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제11권1호
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• pp.23-27
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• 2004

We calculate dim $\hat{A}$ which is a completion of a Noetherian ring A with respect to I-adic topology. We do not use localization but power series techniques.

• 대한수학회보
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• 제55권1호
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• pp.311-317
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• 2018

Let (R, m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module $Hom_R(R/I,H^t_I(R))$ is finitely generated, then $$t={\sup}\{{\dim}{\widehat{\hat{R}_p}}/Q:p{\in}V(I{\hat{R}}),\;Q{\in}mAss{_{\widehat{\hat{R}_p}}}{\widehat{\hat{R}_p}}\;and\;p{\widehat{\hat{R}_p}}=Rad(I{\wideha{\hat{R}_p}}=Q)\}$$. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R{\subset}R[X]$ will be included.