• 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 $S=A{\setminus}(A)$.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.23-32
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• 2023

Let 𝓢 be a Serre class in the category of modules and 𝖆 an ideal of a commutative Noetherian ring R. We study the containment of Tor modules, Koszul homology and local homology in 𝓢 from below. With these results at our disposal, by specializing the Serre class to be Noetherian or zero, a handful of conclusions on Noetherianness and vanishing of the foregoing homology theories are obtained. We also determine when Tor^R_𝓼+t(R/𝖆, X) ≅ Tor^R_𝓼(R/𝖆, H^𝖆_t(X)).

• 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 $$Ext^i_{R[x^S]}(M[x^{-S}],\;N[x^{-S}]){\cong}Ext^i_R(M,\;N)[[x^S]]$$.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.273-280
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• 2024

Let I be an ideal of a commutative Noetherian semi-local ring R and M be an R-module. It is shown that if dim M ≤ 2 and Supp_R M ⊆ V (I), then M is I-weakly cofinite if (and only if) the R-modules Hom_R(R/I, M) and Ext¹_R(R/I, M) are weakly Laskerian. As a consequence of this result, it is shown that the category of all I-weakly cofinite modules X with dim X ≤ 2, forms an Abelian subcategory of the category of all R-modules. Finally, it is shown that if dim R/I ≤ 2, then for each pair of finitely generated R-modules M and N and each pair of the integers i, j ≥ 0, the R-modules Tor^R_i(N, H^j_I(M)) and Extⁱ_R(N, H^j_I(M)) are I-weakly cofinite.

• 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)).

• Honam Mathematical Journal
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• v.23 no.1
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• pp.1-4
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• 2001

In [2, 7.3.2], the set of attached prime ideals of local cohomology module $H_m^n(M)$ were calculated, where (A, m) be Noetherian local ring, M finite A-module and $dim_A(M)=n$, and also in the special case in which furthermore A is a homomorphic image of a Gornestien local ring (A', m') (see [2, 11.3.6]). In this paper, we shall obtain this set, by another way in this special case.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.499-509
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• 2008

In [4] we showed that a polynomial over a Noetherian ring is divisible by some other polynomial by looking at the matrix formed by the coefficients of the polynomials which we called the resultant matrix. Using the result, we will find conditions for a polynomial over a commutative ring to be irreducible. This can be viewed as a generalization of the Eisenstein's irreducibility criterion.

• 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.