• Title, Summary, Keyword: Noetherian ring

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A Note on S-Noetherian Domains

  • LIM, JUNG WOOK
    • Kyungpook Mathematical Journal
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    • v.55 no.3
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    • pp.507-514
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    • 2015
  • Let D be an integral domain, t be the so-called t-operation on D, and S be a (not necessarily saturated) multiplicative subset of D. In this paper, we study the Nagata ring of S-Noetherian domains and locally S-Noetherian domains. We also investigate the t-Nagata ring of t-locally S-Noetherian domains. In fact, we show that if S is an anti-archimedean subset of D, then D is an S-Noetherian domain (respectively, locally S-Noetherian domain) if and only if the Nagata ring $D[X]_N$ is an S-Noetherian domain (respectively, locally S-Noetherian domain). We also prove that if S is an anti-archimedean subset of D, then D is a t-locally S-Noetherian domain if and only if the polynomial ring D[X] is a t-locally S-Noetherian domain, if and only if the t-Nagata ring $D[X]_{N_v}$ is a t-locally S-Noetherian domain.

PROPERTIES OF HURWITZ POLYNOMIAL AND HURWITZ SERIES RINGS

  • Elliott, Jesse;Kim, Hwankoo
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.3
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    • pp.837-849
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    • 2018
  • In this paper, we study the closedness such as seminomality and t-closedness, and Noetherian-like properties such as piecewise Noetherianness and Noetherian spectrum, of Hurwitz polynomial rings and Hurwitz series rings. To do so, we construct an isomorphism between a Hurwitz polynomial ring (resp., a Hurwitz series ring) and a factor ring of a polynomial ring (resp., a power series ring) in a countably infinite number of indeterminates.

REMARKS ON A GOLDBACH PROPERTY

  • Jang, Sun Ju
    • Korean Journal of Mathematics
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    • v.19 no.4
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    • pp.403-407
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    • 2011
  • In this paper, we study Noetherian Boolean rings. We show that if R is a Noetherian Boolean ring, then R is finite and $R{\simeq}(\mathbb{Z}_2)^n$ for some integer $n{\geq}1$. If R is a Noetherian ring, then R/J is a Noetherian Boolean ring, where J is the intersection of all ideals I of R with |R/I| = 2. Thus R/J is finite, and hence the set of ideals I of R with |R/I| = 2 is finite. We also give a short proof of Hayes's result : For every polynomial $f(x){\in}\mathbb{Z}[x]$ of degree $n{\geq}1$, there are irreducible polynomials $g(x)$ and $h(x)$, each of degree $n$, such that $g(x)+h(x)=f(x)$.

ON GRADED KRULL OVERRINGS OF A GRADED NOETHERIAN DOMAIN

  • Lee, Eun-Kyung;Park, Mi-Hee
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.1
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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 $\leq2$.

CHARACTERIZING ABELIAN GENERALIZED REGULAR RINGS THAT ARE NOETHERIAN

  • Han, Juncheol;Sim, Hyo-Seob
    • 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.

INTEGRAL CLOSURE OF A GRADED NOETHERIAN DOMAIN

  • Park, Chang-Hwan;Park, Mi-Hee
    • Journal of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.449-464
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    • 2011
  • We show that, if R is a graded Noetherian ring and I is a proper ideal of R generated by n homogeneous elements, then any prime ideal of R minimal over I has h-height ${\leq}$ n, and that if R is a graded Noetherian domain with h-dim R ${\leq}$ 2, then the integral closure R' of R is also a graded Noetherian domain with h-dim R' ${\leq}$ 2. We also present a short improved proof of the result that, if R is a graded Noetherian domain, then the integral closure of R is a graded Krull domain.

EAKIN-NAGATA THEOREM FOR COMMUTATIVE RINGS WHOSE REGULAR IDEALS ARE FINITELY GENERATED

  • Chang, Gyu Whan
    • Korean Journal of Mathematics
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    • v.18 no.3
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    • pp.271-275
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    • 2010
  • Let R be a commutative ring with identity, T(R) be the total quotient ring of R, and D be a ring such that $R{\subseteq}D{\subseteq}T(R)$ and D is a finite R-module. In this paper, we show that each regular ideal of R is finitely generated if and only if each regular ideal of D is finitely generated. This is a generalization of the Eakin-Nagata theorem that R is Noetherian if and only if D is Noetherian.

ON TYPES OF NOETHERIAN LOCAL RINGS AND MODULES

  • Lee, Ki-Suk
    • Journal of the Korean Mathematical Society
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    • v.44 no.4
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    • pp.987-995
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    • 2007
  • We investigate some results which concern the types of Noetherian local rings. In particular, we show that if r(Ap) ${\le}$ depth Ap + 1 for each prime ideal p of a quasi-unmixed Noetherian local ring A, then A is Cohen-Macaulay. It is also shown that the Kawasaki conjecture holds when dim A ${\le}$ depth A + 1. At the end, we deal with some analogous results for modules, which are derived from the results studied on rings.

SOME REMARKS ON TYPES OF NOETHERIAN LOCAL RINGS

  • Lee, Kisuk
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.4
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    • pp.625-633
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    • 2014
  • We study some results which concern the types of Noetherian local rings, and improve slightly the previous result: For a complete unmixed (or quasi-unmixed) Noetherian local ring A, we prove that if either $A_p$ is Cohen-Macaulay, or $r(Ap){\leq}depth$ $A_p+1$ for every prime ideal p in A, then A is Cohen-Macaulay. Also, some analogous results for modules are considered.