• 대한수학회보
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• 제61권1호
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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.

• 대한수학회보
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• 제50권2호
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• pp.649-657
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• 2013

Let R be a commutative Noetherian ring, I an ideal of R and T be a non-zero I-cofinite R-module with dim(T) ${\leq}$ 1. In this paper, for any finitely generated R-module N with support in V(I), we show that the R-modules $Ext^i_R$(T,N) are finitely generated for all integers $i{\geq}0$. This immediately implies that if I has dimension one (i.e., dim R/I = 1), then $Ext^i_R$($H^j_I$(M), N) is finitely generated for all integers $i$, $j{\geq}0$, and all finitely generated R-modules M and N, with Supp(N) ${\subseteq}$ V(I).

• 호남수학학술지
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• 제45권3호
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• pp.397-409
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• 2023

Let R be an associative ring with identity and M be a left R-module. In this paper, we define modules that have the property (δ-CE) ((δ-CEE)), these are modules that have a δ-supplement (ample δ-supplements) in every cofinite extension which are generalized version of modules that have the properties (CE) and (CEE) introduced in [6] and so a generalization of Zöschinger's modules with the properties (E) and (EE) given in [23]. We investigate various properties of these modules along with examples. In particular we prove these: (1) a module M has the property (δ-CEE) if and only if every submodule of M has the property (δ-CE); (2) direct summands of a module that has the property (δ-CE) also have the property (δ-CE); (3) each factor module of a module that has the property (δ-CE) also has the property (δ-CE) under a special condition; (4) every module with composition series has the property (δ-CE); (5) over a δ-V -ring a module M has the property (δ-CE) if and only if M is cofinitely injective; (6) a ring R is δ-semiperfect if and only if every left R-module has the property (δ-CE).

• 대한수학회보
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• 제55권1호
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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.