• Title/Summary/Keyword: Noetherian ring

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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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Injective Property Of Generalized Inverse Polynomial Module

  • Park, Sang-Won
    • Communications of the Korean Mathematical Society
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    • v.15 no.2
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    • pp.257-261
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    • 2000
  • Northcott and Mckerrow proved that if R is a left noe-therian ring and E is an injective left R-module, then E[x-1] is an injective left R[x]-module. In this paper we generalize Northcott and McKerrow's result so that if R is a left noetherian ring and E is an in-jective left R-module, then E[x-S] is an injective left R[xS]-module, where S is a submonoid of N (N is the set of all natural numbers).

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ALMOST MULTIPLICATIVE SETS

  • BAEK, HYUNG TAE;LIM, JUNG WOOK
    • Journal of applied mathematics & informatics
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    • v.39 no.3_4
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    • pp.259-266
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    • 2021
  • Let R be a commutative ring with identity and let S be a nonempty subset of R. We define S to be an almost multiplicative subset of R if for each a, b ∈ S, there exist integers m, n ≥ 1 such that ambn ∈ S. In this article, we study some utilization of almost multiplicative subsets.

On Injectivity of Modules via Semisimplicity

  • Nguyen, Thi Thu Ha
    • Kyungpook Mathematical Journal
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    • v.62 no.4
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    • pp.641-655
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    • 2022
  • A right R-module N is called pseudo semisimple-M-injective if for any monomorphism from every semisimple submodule of M to N, can be extended to a homomorphism from M to N. In this paper, we study some properties of pseudo semisimple-injective modules. Moreover, some results of pseudo semisimple-injective modules over formal triangular matrix rings are obtained.

Asymptotic behavior of ideals relative to injective A-modules

  • Song, Yeong-Moo
    • Communications of the Korean Mathematical Society
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    • v.10 no.3
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    • pp.491-498
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    • 1995
  • This paper is concerned with an asymptotic behavior of ideals relative to injective modules ove the commutative Noetherian ring A : under what conditions on A can we show that $$\bar{At^*}(a,E)=At^*(a,E)$?

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SEMISIMPLE DIMENSION OF MODULES

  • Amirsardari, Bahram;Bagheri, Saeid
    • Communications of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.711-719
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    • 2018
  • In this paper we define and study a new kind of dimension called, semisimple dimension, that measures how far a module is from being semisimple. Like other kinds of dimensions, this is an ordinal valued invariant. We give some interesting and useful properties of rings or modules which have semisimple dimension. It is shown that a noetherian module with semisimple dimension is an artinian module. A domain with semisimple dimension is a division ring. Also, for a semiprime right non-singular ring R, if its maximal right quotient ring has semisimple dimension as a right R-module, then R is a semisimple artinian ring. We also characterize rings whose modules have semisimple dimension. In fact, it is shown that all right R-modules have semisimple dimension if and only if the free right R-module ${\oplus}^{\infty}_{i=1}$ R has semisimple dimension, if and only if R is a semisimple artinian ring.

WEAKLY DUO RINGS WITH NIL JACOBSON RADICAL

  • KIM HONG KEE;KIM NAM KYUN;LEE YANG
    • Journal of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.457-470
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    • 2005
  • Yu showed that every right (left) primitive factor ring of weakly right (left) duo rings is a division ring. It is not difficult to show that each weakly right (left) duo ring is abelian and has the classical right (left) quotient ring. In this note we first provide a left duo ring (but not weakly right duo) in spite of it being left Noetherian and local. Thus we observe conditions under which weakly one-sided duo rings may be two-sided. We prove that a weakly one-sided duo ring R is weakly duo under each of the following conditions: (1) R is semilocal with nil Jacobson radical; (2) R is locally finite. Based on the preceding case (1) we study a kind of composition length of a right or left Artinian weakly duo ring R, obtaining that i(R) is finite and $\alpha^{i(R)}R\;=\;R\alpha^{i(R)\;=\;R\alpha^{i(R)}R\;for\;all\;\alpha\;{\in}\;R$, where i(R) is the index (of nilpotency) of R. Note that one-sided Artinian rings and locally finite rings are strongly $\pi-regular$. Thus we also observe connections between strongly $\pi-regular$ weakly right duo rings and related rings, constructing available examples.

m-PRIMARY m-FULL IDEALS

  • Woo, Tae Whan
    • Journal of the Chungcheong Mathematical Society
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    • v.26 no.4
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    • pp.799-809
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    • 2013
  • An ideal I of a local ring (R, m, k) is said to be m-full if there exists an element $x{\in}m$ such that Im : x = I. An ideal I of a local ring R is said to have the Rees property if ${\mu}$(I) > ${\mu}$(J) for any ideal J containing I. We study properties of m-full ideals and we characterize m-primary m-full ideals in terms of the minimal number of generators of the ideals. In particular, for a m-primary ideal I of a 2-dimensional regular local ring (R, m, k), we will show that the following conditions are equivalent. 1. I is m-full 2. I has the Rees property 3. ${\mu}$(I)=o(I)+1 In this paper, let (R, m, k) be a commutative Noetherian local ring with infinite residue field k = R/m.

COUSIN COMPLEXES AND GENERALIZED HUGHES COMPLEXES

  • Kim, Dae-Sig;Song, Yeong-Moo
    • Communications of the Korean Mathematical Society
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    • v.9 no.3
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    • pp.503-511
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    • 1994
  • In this paper, the ring A will mean a commutative Noetherian ring with non-zero multiplicative identity, it is understood that the ring homomorphisms respect identity elements and M will denote an A-module. Throughout this paper A and B will denote rings, $f : A \to B$ a ring homomorphism. C(A) (resp. C(B)) presents the category of all A-modules (resp. B-modules) and A-homomorphisms (resp. B-homorphisms) between them. The following ideas will be used without further explanation. B can be regarded as an A-module by means of f and $M\otimesB$ can be regarded as a B-module in the natural way. Furthermore the restriction of scalars provides a functor from C(B) to C(A).

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