• Title, Summary, Keyword: Noetherian ring

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A NOTE ON TYPES OF NOETHERIAN LOCAL RINGS

  • Lee, Kisuk
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.4
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    • pp.645-652
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    • 2002
  • In this note we investigate some results which concern the types of local rings. In particular it is shown that if the type of a quasi-unmixed local ring A is less than or equal to depth A + 1, and $\hat{A}_p$ is Cohen-Macaulay for every prime $p\neq\hat{m}$, then A is Cohen-Macaulay. (This implies the previously known result: if A satisfies $(S_{n-1})}$, where n is the type of a .ins A, then A is Cohen-Macaulay.)

FINITELY GENERATED PROJECTIVE MODULES OVER NOETHERIAN RINGS

  • LEE, SANG CHEOL;KIM, SUNAH
    • Honam Mathematical Journal
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    • v.28 no.4
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    • pp.499-511
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    • 2006
  • It is well-known that every finitely generated torsion-free module over a principal ideal domain is free. This will be generalized. We deal with ideals of the finite, external direct product of certain rings. Finally, if M is a torsion-free, finitely generated module over a reduced, Noetherian ring A, then we prove that Ms is a projective module over As, where $S=A{\setminus}(A)$.

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A Characterization of Dedekind Domains and ZPI-rings

  • Rostami, Esmaeil
    • Kyungpook Mathematical Journal
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    • v.57 no.3
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    • pp.433-439
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    • 2017
  • It is well known that an integral domain D is a Dedekind domain if and only if D is a Noetherian almost Dedekind domain. In this paper, we show that an integral domain D is a Dedekind domain if and only if D is an almost Dedekind domain such that Max(D) is a Noetherian topological space as a subspace of Spec(D) with respect to the Zariski topology. We also give a new characterization of ZPI-rings.

HOM AND EXT FUNCTORS OF GENERALIZED INVERSE POLYNOMIAL MODULES

  • Han, Chang-Woo;Park, Sang-Won;Cho, Eun-Ha
    • East Asian mathematical journal
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    • v.16 no.1
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    • pp.111-123
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    • 2000
  • Northcott and McKerrow proved that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-1}]$ is an injective left R[xl-module. Park generalize Northcott and McKerrow's result so that if R is a left noetherian ring and E is an injective left R-module, then $E[x^{-S}]$ is an injective left $R[x^s]$-module, where S is a submonoid of N(N is the set of all natural numbers). In this paper we show $$Hom_{R[x^S]}(M[x^{-S}],\;N[x^{-S}]){\cong}Hom_R(M,\;N)[[x^S]]$$ and using the above result and this isomorphism, finally we show that $$Ext^i_{R[x^S]}(M[x^{-S}],\;N[x^{-S}]){\cong}Ext^i_R(M,\;N)[[x^S]]$$.

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ON THE COHOMOLOGICAL DIMENSION OF FINITELY GENERATED MODULES

  • Bahmanpour, Kamal;Samani, Masoud Seidali
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.311-317
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    • 2018
  • Let (R, m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module $Hom_R(R/I,H^t_I(R))$ is finitely generated, then $$t={\sup}\{{\dim}{\widehat{\hat{R}_p}}/Q:p{\in}V(I{\hat{R}}),\;Q{\in}mAss{_{\widehat{\hat{R}_p}}}{\widehat{\hat{R}_p}}\;and\;p{\widehat{\hat{R}_p}}=Rad(I{\wideha{\hat{R}_p}}=Q)\}$$. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R{\subset}R[X]$ will be included.

SOME CHARACTERIZATIONS OF COHEN-MACAULAY MODULES IN DIMENSION > s

  • Dung, Nguyen Thi
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.519-530
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    • 2014
  • Let (R,m) be a Noetherian local ring and M a finitely generated R-module. For an integer s > -1, we say that M is Cohen-Macaulay in dimension > s if every system of parameters of M is an M-sequence in dimension > s introduced by Brodmann-Nhan [1]. In this paper, we give some characterizations for Cohen-Macaulay modules in dimension > s in terms of the Noetherian dimension of the local cohomology modules $H^i_m(M)$, the polynomial type of M introduced by Cuong [5] and the multiplicity e($\underline{x}$;M) of M with respect to a system of parameters $\underline{x}$.

COPRIMELY PACKED RINGS (II)

  • CHO, YONG HWAN
    • Honam Mathematical Journal
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    • v.21 no.1
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    • pp.43-47
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    • 1999
  • In this paper we show that in case of Noetherian ring R(1) if R is coprimely packed then R((X)) is coprimely packed and (2) if Max(R) is coprimely packed then so is MaxR((X)).

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A NOTE OF PI-RINGS WITH RESTRICTED DESCENDING

  • Hong, Chan-Yong
    • The Pure and Applied Mathematics
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    • v.1 no.1
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    • pp.1-6
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    • 1994
  • In this paper, some properties for a PI-ring satisfying the descending chain condition on essential left ideals are studied: Let R be a ring with a polynomial identity satisfying the descending chain condition on essential ideals. Then all minimal prime ideals in R are maximal ideals. Moreover, if R has only finitely many minimal prime ideals, then R is left and right Artinian. Consequently, if every primeideal of R is finitely generated as a left ideal, then R is left and right Artinian. A finitely generated PI-algebra over a commutative Noetherian ring satisfying the descending chain condition on essential left ideals is a finite module over its center.(omitted)

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Some Properties of Generalized Fractions

  • Lee, Dong-Soo;Chung, Sang-Cho
    • Journal of the Chungcheong Mathematical Society
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    • v.7 no.1
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    • pp.153-164
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    • 1994
  • Let A be a commutative ring with identity and M an A-module. When $U_n$ is a triangular subset of $A_n$, Sharp and Zakeri defined a module of generalized fractions $U_n^{-n}M$. In [SZ3], they described a relation of the Monomial Conjecture and a module of generalized fractions under the condition of a Noetherian local ring. In this paper, we investigate some properties of non-zero generalized fractions and give a generalization of results of Sharp and Zakeri for an arbitrary ring.

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ON SUBSTRUCTURES OF MONOGENIC R-GROUPS

  • Cho, Yong-Uk
    • Journal of applied mathematics & informatics
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    • v.26 no.1_2
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    • pp.401-406
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    • 2008
  • In this paper, we will introduce the noetherian quotients in R-groups, and then investigate the related substructures of the near-ring R and G and the R-group G. Also, applying the annihilator concept in R-groups and d.g. near-rings, we will survey some properties of the substructures of R and G in monogenic Rgroups, and show that R becomes a ring for faithful monogenic R-groups with some condition.

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