• Title, Summary, Keyword: Noetherian ring

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PROPERTIES OF NOETHERIAN QUOTIENTS IN R-GROUPS

  • Cho, Yong Uk
    • Journal of the Chungcheong Mathematical Society
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    • v.20 no.2
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    • pp.183-190
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    • 2007
  • 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 R-groups and faithful R-groups.

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A NOTE ON THE LOCAL HOMOLOGY

  • Rasoulyar, S.
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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    • pp.387-391
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    • 2004
  • Let A be Noetherian ring, a= (${\tau}_1..., \tau_n$ an ideal of A and $C_{A}$ be category of A-modules and A-homomorphisms. We show that the connected left sequences of covariant functors ${limH_i(K.(t^t,-))}_{i\geq0}$ and ${lim{{Tor^A}_i}(\frac{A}{a^f}-)}_{i\geq0}$ are isomorphic from $C_A$ to itself, where $\tau^t\;=\;{{\tau_^t}_1$, ㆍㆍㆍ${\tau^t}_n$.

ON THE PRIME SPECTRUM OF A MODULE OVER A COMMUTATIVE NOETHERIAN RING

  • Ansari-Toroghy, H.;Sarmazdeh-Ovlyaee, R.
    • Honam Mathematical Journal
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    • v.29 no.3
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    • pp.351-366
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    • 2007
  • Let R be a commutative ring and let M be an R-module. Let X = Spec(M) be the prime spectrum of M with Zariski topology. Our main purpose in this paper is to specify the topological dimensions of X, where X is a Noetherian topological space, and compare them with those of topological dimensions of $Supp_{R}$(M). Also we will give a characterization for the irreducibility of X and we obtain some related results.

ASYMPTOTIC BEHAVIOUR OF IDEALS RELATIVE TO SOME MODULES OVER A COMMUTATIVE NOETHERIAN RING

  • ANSARI-TOROGHY, H.
    • Honam Mathematical Journal
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    • v.23 no.1
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    • pp.5-14
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    • 2001
  • Let E be an injective module over a commutative Noetherian ring A. In this paper we will show that if I is regular ideal, then the sequence of sets $$Ass_A((I^n)^{{\star}(E)}/I^n),\;n{\in}N$$ is ultimately constant. Also we obtain some related results. (Here for an ideal J of A, $J^{{\star}(E)}$ denotes the integral closure of J relative to E.

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ON 𝜙-PSEUDO-KRULL RINGS

  • El Khalfi, Abdelhaq;Kim, Hwankoo;Mahdou, Najib
    • Communications of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.1095-1106
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    • 2020
  • The purpose of this paper is to introduce a new class of rings that is closely related to the class of pseudo-Krull domains. Let 𝓗 = {R | R is a commutative ring and Nil(R) is a divided prime ideal of R}. Let R ∈ 𝓗 be a ring with total quotient ring T(R) and define 𝜙 : T(R) → RNil(R) by ${\phi}({\frac{a}{b}})={\frac{a}{b}}$ for any a ∈ R and any regular element b of R. Then 𝜙 is a ring homomorphism from T(R) into RNil(R) and 𝜙 restricted to R is also a ring homomorphism from R into RNil(R) given by ${\phi}(x)={\frac{x}{1}}$ for every x ∈ R. We say that R is a 𝜙-pseudo-Krull ring if 𝜙(R) = ∩ Ri, where each Ri is a nonnil-Noetherian 𝜙-pseudo valuation overring of 𝜙(R) and for every non-nilpotent element x ∈ R, 𝜙(x) is a unit in all but finitely many Ri. We show that the theories of 𝜙-pseudo Krull rings resemble those of pseudo-Krull domains.

Completely Indecomposable Modules over a Ring

  • Kim, Sunah;Park, Soon-Chul
    • Honam Mathematical Journal
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    • v.3 no.1
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    • pp.109-113
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    • 1981
  • 본(本) 논문(論文)에서는 Noetherian Ring 상(上)의 finitely generated injective module이 completely indecomposable modules의 direct sum으로 표시(表示)될 필요충분조건(必要充分條件)을 구(求)하였다.

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LOCALLY COMPLETE INTERSECTION IDEALS IN COHEN-MACAULAY LOCAL RINGS

  • Kim, Mee-Kyoung
    • Communications of the Korean Mathematical Society
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    • v.9 no.2
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    • pp.261-264
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    • 1994
  • Throughout this paper, all rings are assumed to be commutative with identity. By a local ring (R, m), we mean a Noetherian ring R which has the unique maximal ideal m. By dim(R) we always mean the Krull dimension of R. Let I be an ideal in a ring R and t an indeterminate over R. Then the Rees algebra R[It] is defined to be(omitted)

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INJECTIVE PROPERTY OF LAURENT POWER SERIES MODULE

  • Park, Sang-Won
    • East Asian mathematical journal
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    • v.17 no.2
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    • pp.367-374
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    • 2001
  • 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[x]-module. Park generalized 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 $\mathbb{N}$($\mathbb{N}$ is the set of all natural numbers). In this paper we extend the injective property to the Laurent power series module so that if R is a ring and E is an injective left R-module, then $E[[x^{-1},x]]$ is an injective left $R[x^S]$-module.

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