• Title, Summary, Keyword: Noetherian ring

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ON PIECEWISE NOETHERIAN DOMAINS

  • Chang, Gyu Whan;Kim, Hwankoo;Wang, Fanggui
    • Journal of the Korean Mathematical Society
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    • v.53 no.3
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    • pp.623-643
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    • 2016
  • In this paper, we study piecewise Noetherian (resp., piecewise w-Noetherian) properties in several settings including flat (resp., t-flat) overrings, Nagata rings, integral domains of finite character (resp., w-finite character), pullbacks of a certain type, polynomial rings, and D + XK[X] constructions.

INJECTIVE MODULES OVER ω-NOETHERIAN RINGS, II

  • Zhang, Jun;Wang, Fanggui;Kim, Hwankoo
    • Journal of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1051-1066
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    • 2013
  • By utilizing known characterizations of ${\omega}$-Noetherian rings in terms of injective modules, we give more characterizations of ${\omega}$-Noetherian rings. More precisely, we show that a commutative ring R is ${\omega}$-Noetherian if and only if the direct limit of GV -torsion-free injective R-modules is injective; if and only if every R-module has a GV -torsion-free injective (pre)cover; if and only if the direct sum of injective envelopes of ${\omega}$-simple R-modules is injective; if and only if the essential extension of the direct sum of GV -torsion-free injective R-modules is the direct sum of GV -torsion-free injective R-modules; if and only if every $\mathfrak{F}_{w,f}(R)$-injective ${\omega}$-module is injective; if and only if every GV-torsion-free R-module admits an $i$-decomposition.

ON A CHANGE OF RINGS FOR MIXED MULTIPLICITIES

  • Thanh, Truong Thi Hong;Viet, Duong Quoc
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1251-1258
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    • 2020
  • This paper establishes a formula changing the ring from a Noetherian local ring A of dimension d > 0 containing the residue field k to the polynomial ring in d variables k[X1, X2, …, Xd] for mixed multiplicities. And as consequences, we get a formula for the multiplicity of Rees rings and formulas for mixed multiplicities and the multiplicity of Rees rings of quotient rings of A by highest dimensional associated prime ideals of A.

MATLIS INJECTIVE MODULES

  • Yan, Hangyu
    • Bulletin of the Korean Mathematical Society
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    • v.50 no.2
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    • pp.459-467
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    • 2013
  • In this paper, Matlis injective modules are introduced and studied. It is shown that every R-module has a (special) Matlis injective preenvelope over any ring R and every right R-module has a Matlis injective envelope when R is a right Noetherian ring. Moreover, it is shown that every right R-module has an ${\mathcal{F}}^{{\perp}1}$-envelope when R is a right Noetherian ring and $\mathcal{F}$ is a class of injective right R-modules.

KRULL DIMENSION OF A COMPLETION

  • Hwnag, Chul-Ju
    • 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.

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