• Title, Summary, Keyword: Noetherian ring

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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.

REDUCTIONS OF IDEALS IN COMMUTATIVE NOETHERIAN SEMI-LOCAL RINGS

  • Song, Yeong-Moo;Kim, Se-Gyeong
    • Communications of the Korean Mathematical Society
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    • v.11 no.3
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    • pp.539-546
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    • 1996
  • The purpose of this paper is to show that the Noetherian semi-local property of the underlying ring enables us to develope a setisfactory concep of the theory of reduction of ideals in a commutative Noetherian ring.

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MINIMAL PROJECTIVE RESOLUTIONS OF A FINITELY GENERATED MODULE M OVER A NOETHERIAN LOCAL RING (R, 𝔪) AND THE COHOMOLOGIES OF (M, R/𝔪)

  • Lee, Sang Cheol;Song, Yeong Moo
    • Honam Mathematical Journal
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    • v.40 no.2
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    • pp.355-366
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    • 2018
  • Let R be a commutative ring with identity and let M be a finitely generated module over a Noetherian local ring R. Then it is well-known that M has a minimal projective resolution, which is unique up to isomorphisms of exact sequences. We provide a new proof of its uniqueness. Moreover, we deal with the cohomologies of (M, R/m).

A NOTE ON w-NOETHERIAN RINGS

  • Xing, Shiqi;Wang, Fanggui
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.2
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    • pp.541-548
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    • 2015
  • Let R be a commutative ring. An R-module M is called a w-Noetherian module if every submodule of M is of w-finite type. R is called a w-Noetherian ring if R as an R-module is a w-Noetherian module. In this paper, we present an exact version of the Eakin-Nagata Theorem on w-Noetherian rings. To do this, we prove the Formanek Theorem for w-Noetherian rings. Further, we point out by an example that the condition (${\dag}$) in the Chung-Ha-Kim version of the Eakin-Nagata Theorem on SM domains is essential.

HILBERT BASIS THEOREM FOR RINGS WITH ∗-NOETHERIAN SPECTRUM

  • PARK, MIN JI;LIM, JUNG WOOK
    • Journal of applied mathematics & informatics
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    • v.38 no.3_4
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    • pp.271-276
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    • 2020
  • Let R be a commutative ring with identity, R[X] the polynomial ring over R, ∗ a radical operation on R and ⋆ a radical operation of finite character on R[X]. In this paper, we give Hilbert basis theorem for rings with ∗-Noetherian spectrum. More precisely, we show that if (IR[X]) = (IR[X]) and (IR[X]) ∩ R = I for all ideals I of R, then R has ∗-Noetherian spectrum if and only if R[X] has ⋆-Noetherian spectrum. This is a generalization of a well-known fact that R has Noetherian spectrum if and only if R[X] has Noetherian spectrum.

GENERALIZED QUASI-PRIMARY RINGS

  • MOGHIMI, HOSEIN FAZAELI;SAMIEI, MAHDI
    • Advanced Studies in Contemporary Mathematics
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    • v.28 no.4
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    • pp.607-613
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    • 2018
  • In this paper, the structure of commutative rings with identity all of whose ideals a re quasi-primary, called generalized quasi-primary rings, is studied and several equivalent conditions to such rings are considered. Equivalently, a generalized quasi-primary ring may be viewed as a ring whose the set of radical ideals forms a chain. It is proved that an Artinian local ring R is a generalized quasi-primary ring and the converse is true if R is a non-domain Noetherian ring.

A NOTE ON ZERO DIVISORS IN w-NOETHERIAN-LIKE RINGS

  • Kim, Hwankoo;Kwon, Tae In;Rhee, Min Surp
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.6
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    • pp.1851-1861
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    • 2014
  • We introduce the concept of w-zero-divisor (w-ZD) rings and study its related rings. In particular it is shown that an integral domain R is an SM domain if and only if R is a w-locally Noetherian w-ZD ring and that a commutative ring R is w-Noetherian if and only if the polynomial ring in one indeterminate R[X] is a w-ZD ring. Finally we characterize universally zero divisor rings in terms of w-ZD modules.

A Note on Gaussian Series Rings

  • Kim, Eun Sup;Lee, Seung Min;Lim, Jung Wook
    • Kyungpook Mathematical Journal
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    • v.57 no.3
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    • pp.419-431
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    • 2017
  • In this paper, we define a new kind of formal power series rings by using Gaussian binomial coefficients and investigate some properties. More precisely, we call such a ring a Gaussian series ring and study McCoy's theorem, Hermite properties and Noetherian properties.

*-NOETHERIAN DOMAINS AND THE RING D[X]N*, II

  • Chang, Gyu-Whan
    • Journal of the Korean Mathematical Society
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    • v.48 no.1
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    • pp.49-61
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    • 2011
  • Let D be an integral domain with quotient field K, X be a nonempty set of indeterminates over D, * be a star operation on D, $N_*$={f $\in$ D[X]|c(f)$^*$= D}, $*_w$ be the star operation on D defined by $I^{*_w}$ = ID[X]${_N}_*$ $\cap$ K, and [*] be the star operation on D[X] canonically associated to * as in Theorem 2.1. Let $A^g$ (resp., $A^{[*]g}$, $A^{[*]g}$) be the global (resp.,*-global, [*]-global) transform of a ring A. We show that D is a $*_w$-Noetherian domain if and only if D[X] is a [*]-Noetherian domain. We prove that $D^{*g}$[X]${_N}_*$ = (D[X]${_N}_*$)$^g$ = (D[X])$^{[*]g}$; hence if D is a $*_w$-Noetherian domain, then each ring between D[X]${_N}_*$ and $D^{*g}$[X]${_N}_*$ is a Noetherian domain. Let $\tilde{D}$ = $\cap${$D_P$|P $\in$ $*_w$-Max(D) and htP $\geq$2}. We show that $D\;\subseteq\;\tilde{D}\;\subseteq\;D^{*g}$ and study some properties of $\tilde{D}$ and $D^{*g}$.

ON SOME GENERALIZATIONS OF CLOSED SUBMODULES

  • DURGUN, YILMAZ
    • 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.