• Title, Summary, Keyword: Noetherian ring

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ω-MODULES OVER COMMUTATIVE RINGS

  • Yin, Huayu;Wang, Fanggui;Zhu, Xiaosheng;Chen, Youhua
    • Journal of the Korean Mathematical Society
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    • v.48 no.1
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    • pp.207-222
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    • 2011
  • Let R be a commutative ring and let M be a GV -torsionfree R-module. Then M is said to be a $\omega$-module if $Ext_R^1$(R/J, M) = 0 for any J $\in$ GV (R), and the w-envelope of M is defined by $M_{\omega}$ = {x $\in$ E(M) | Jx $\subseteq$ M for some J $\in$ GV (R)}. In this paper, $\omega$-modules over commutative rings are considered, and the theory of $\omega$-operations is developed for arbitrary commutative rings. As applications, we give some characterizations of $\omega$-Noetherian rings and Krull rings.

STRUCTURE OF THE FLAT COVERS OF ARTINIAN MODULES

  • Payrovi, S.H.
    • Journal of the Korean Mathematical Society
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    • v.39 no.4
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    • pp.611-620
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    • 2002
  • The aim of the Paper is to Obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let R be a commutative Noetherian ring and let A be an Artinian R-module. We prove that the flat cover of a is of the form $\prod_{p\epsilonAtt_R(A)}T-p$, where $Tp$ is the completion of a free R$_{p}$-module. Also, we construct a minimal flat resolution for R/xR-module 0: $_AX$ from a given minimal flat resolution of A, when n is a non-unit and non-zero divisor of R such that A = $\chiA$. This result leads to a description of the structure of a minimal flat resolution for ${H^n}_{\underline{m}}(R)$, nth local cohomology module of R with respect to the ideal $\underline{m}$, over a local Cohen-Macaulay ring (R, $\underline{m}$) of dimension n.

A NOTE ON COHOMOLOGICAL DIMENSION OVER COHEN-MACAULAY RINGS

  • Bagheriyeh, Iraj;Bahmanpour, Kamal;Ghasemi, Ghader
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.275-280
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    • 2020
  • Let (R, m) be a Noetherian local Cohen-Macaulay ring and I be a proper ideal of R. Assume that βR(I, R) denotes the constant value of depthR(R/In) for n ≫ 0. In this paper we introduce the new notion γR(I, R) and then we prove the following inequalities: βR(I, R) ≤ γR(I, R) ≤ dim R - cd(I, R) ≤ dim R/I. Also, some applications of these inequalities will be included.

SOME RESULTS ON INTEGER-VALUED POLYNOMIALS OVER MODULES

  • Naghipour, Ali Reza;Hafshejani, Javad Sedighi
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.5
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    • pp.1165-1176
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    • 2020
  • Let M be a module over a commutative ring R. In this paper, we study Int(R, M), the module of integer-valued polynomials on M over R, and IntM(R), the ring of integer-valued polynomials on R over M. We establish some properties of Krull dimensions of Int(R, M) and IntM(R). We also determine when Int(R, M) and IntM(R) are nontrivial. Among the other results, it is shown that Int(ℤ, M) is not Noetherian module over IntM(ℤ) ∩ Int(ℤ), where M is a finitely generated ℤ-module.

QUASI-COMPLETENESS AND LOCALIZATIONS OF POLYNOMIAL DOMAINS: A CONJECTURE FROM "OPEN PROBLEMS IN COMMUTATIVE RING THEORY"

  • Farley, Jonathan David
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.6
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    • pp.1613-1615
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    • 2016
  • It is proved that $k[X_1,{\ldots},X_v ]$ localized at the ideal ($X_1,{\ldots},X_v$ ), where k is a field and $X_1,{\ldots},X_v$ indeterminates, is not weakly quasi-complete for $v{\geq}2$, thus proving a conjecture of D. D. Anderson and solving a problem from "Open Problems in Commutative Ring Theory" by Cahen, Fontana, Frisch, and Glaz.

SOME NEW CHARACTERIZATIONS OF QUASI-FROBENIUS RINGS BY USING PURE-INJECTIVITY

  • Moradzadeh-Dehkordi, Ali
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.371-381
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    • 2020
  • A ring R is called right pure-injective if it is injective with respect to pure exact sequences. According to a well known result of L. Melkersson, every commutative Artinian ring is pure-injective, but the converse is not true, even if R is a commutative Noetherian local ring. In this paper, a series of conditions under which right pure-injective rings are either right Artinian rings or quasi-Frobenius rings are given. Also, some of our results extend previously known results for quasi-Frobenius rings.

Bipolar fuzzy ideals of Near Rings

  • Baik, Hyoung-Gu
    • Journal of the Korean Institute of Intelligent Systems
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    • v.22 no.3
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    • pp.394-398
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    • 2012
  • Based on the theory of a bipolar fuzzy set, the notion of a bipolar fuzzy subring/ideal of a Near ring is introduced and related properties are investigated. Characterizations of a bipolar fuzzy subnear ring and a bipolar fuzzy ideal in near ring are established. Relations between a bipolar fuzzy ideal and a level cut are discussed. Using bipolar fuzzy ideals, we discuss characterizations of Noetherian Near ring.

EXACTNESS OF IDEAL TRANSFORMS AND ANNIHILATORS OF TOP LOCAL COHOMOLOGY MODULES

  • BAHMANPOUR, KAMAL
    • Journal of the Korean Mathematical Society
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    • v.52 no.6
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    • pp.1253-1270
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    • 2015
  • Let (R, m) be a commutative Noetherian local domain, M a non-zero finitely generated R-module of dimension n > 0 and I be an ideal of R. In this paper it is shown that if $x_1,{\ldots },x_t$ ($1{\leq}t{\leq}n$) be a sub-set of a system of parameters for M, then the R-module $H^t_{(x_1,{\ldots },x_t)}$(R) is faithful, i.e., Ann $H^t_{(x_1,{\ldots },x_t)}$(R) = 0. Also, it is shown that, if $H^i_I$ (R) = 0 for all i > dim R - dim R/I, then the R-module $H^{dimR-dimR/I}_I(R)$ is faithful. These results provide some partially affirmative answers to the Lynch's conjecture in [10]. Moreover, for an ideal I of an arbitrary Noetherian ring R, we calculate the annihilator of the top local cohomology module $H^1_I(M)$, when $H^i_I(M)=0$ for all integers i > 1. Also, for such ideals we show that the finitely generated R-algebra $D_I(R)$ is a flat R-algebra.

ON FUZZY QUOTIENT RINGS AND CHAIN CONDITIONS

  • Lee, Kyoung-Hee
    • The Pure and Applied Mathematics
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    • v.7 no.1
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    • pp.33-40
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    • 2000
  • We prove some characterization of rings with chain conditions in terms of fuzzy quotient rings and fuzzy ideals. We also show that a ring R is left Artinian if and only of the set of values of every fuzzy ideal on R is upper well-ordered.

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