• Journal of the Korean Mathematical Society
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• v.50 no.5
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• pp.991-1008
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• 2013

In this paper, we mainly discuss Gorenstein Dedekind do-mains (G-Dedekind domains for short) and their overrings. Let R be a one-dimensional Noetherian domain with quotient field K and integral closure T. Then it is proved that R is a G-Dedekind domain if and only if for any prime ideal P of R which contains ($R\;:_K\;T$), P is Gorenstein projective. We also give not only an example to show that G-Dedekind domains are not necessarily Noetherian Warfield domains, but also a definition for a special kind of domain: a 2-DVR. As an application, we prove that a Noetherian domain R is a Warfield domain if and only if for any maximal ideal M of R, $R_M$ is a 2-DVR.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1549-1557
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• 2015

Characterizations of closed subgroups in abelian groups have been generalized to modules in essentially dierent ways; they are in general inequivalent. Here we consider the relations between these generalizations over commutative rings, and we characterize the commutative rings over which they coincide. These are exactly the commutative noetherian distributive rings. We also give a characterization of c-injective modules over commutative noetherian distributive rings. For a noetherian distributive ring R, we prove that, (1) direct product of simple R-modules is c-injective; (2) an R-module D is c-injective if and only if it is isomorphic to a direct summand of a direct product of simple R-modules and injective R-modules.

• East Asian mathematical journal
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• v.36 no.1
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• pp.73-79
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• 2020

A ring R is called generalized regular if for every nonzero x in R there exists y in R such that xy is a nonzero idempotent. In this paper, we observe some equivalent conditions for the generalized regular rings that are abelian in terms of idempotents, and we also investigate the primitivity of an idempotent for such a ring. By using the investigation, we characterize such a kind of rings that are noetherian by showing that an abelian generalized regular ring R is noetherian if and only if R is isomorphic to a direct product of finitely many division rings. We also observe some interesting consequences of our results.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.617-622
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• 2021

Let R be a commutative ring with identity, R[X] the polynomial ring over R and S a multiplicative subset of R. Let U = {f ∈ R[X] | f is monic} and let N = {f ∈ R[X] | c(f) = R}. In this paper, we show that if S is an anti-Archimedean subset of R, then R is an S-Noetherian ring if and only if R[X]_U is an S-Noetherian ring, if and only if R[X]_N is an S-Noetherian ring. We also prove that if R is an integral domain and R[X]_U is an S-principal ideal domain, then R is an S-principal ideal domain.

• The Pure and Applied Mathematics
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• v.11 no.1
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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.

• The Pure and Applied Mathematics
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• v.2 no.1
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• pp.61-65
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• 1995

In this paper we introduce the notion of Artinian and obtain some properties of Artinian and Noetherian BCK-algebras.

• Honam Mathematical Journal
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• v.40 no.2
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• pp.219-225
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• 2018

In this paper, we characterize the prime submodules of an injective module over a Noetherian ring.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.45-51
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• 2018

This paper contains some results that grew out of an attempt to Camillo-Krause conjecture: Is a ring R right Noetherian if for each nonzero right ideal I of R, R/I is an Artinian right R-module?

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.103-110
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• 2019

In this paper, we give a characterization of prime submodules of an injective module over a Noetherian ring.

• 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.