• Korean Journal of Mathematics
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• v.28 no.1
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• pp.65-73
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• 2020

This article concerns a property of local rings and domains. A ring R is called weakly local if for every a ∈ R, a is regular or 1-a is regular, where a regular element means a non-zero-divisor. We study the structure of weakly local rings in relation to several kinds of factor rings and ring extensions that play roles in ring theory. We prove that the characteristic of a weakly local ring is either zero or a power of a prime number. It is also shown that the weakly local property can go up to polynomial (power series) rings and a kind of Abelian matrix rings.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.637-647
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• 2023

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, M an arbitrary R-module and X a finite R-module. We prove a characterization for Hⁱ_𝔞(M) and Hⁱ_𝔞(X, M) to be 𝔞-weakly cofinite for all i, whenever one of the following cases holds: (a) ara(𝔞) ≤ 1, (b) dim R/𝔞 ≤ 1 or (c) dim R ≤ 2. We also prove that, if M is a weakly Laskerian R-module, then Hⁱ_𝔞(X, M) is 𝔞-weakly cofinite for all i, whenever dim X ≤ 2 or dim M ≤ 2 (resp. (R, m) a local ring and dim X ≤ 3 or dim M ≤ 3). Let d = dim M < ∞, we prove an equivalent condition for top local cohomology module H^d_𝔞(M) to be weakly Artinian.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.28 no.2
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• pp.175-180
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• 1991

In [3], [4] and [5], Hochster and Huneke introduced the notions of the tight closure of an ideal and of the weak F-regularity of a ring. This notion enabled us to give new proofs of many results in commutative algebra. A regular ring is known to be F-regular, and a Gorenstein local ring is proved to be F-regular provided that one ideal generated by a system of parameters (briefly s.o.p.) is tightly closed. In fact, a Gorenstein local ring is weakly F-regular if and only if there exists a system of parameters ideal which is tightly closed [3]. But we do not know whether this fact is true or not if a ring is not Gorenstein, in particular, a ring is a Cohen Macaulay (briefly C-M) local ring. In this paper, we will prove this in the case of an 1-dimensional C-M local ring. For this, we study the F-rationality and the normality of the ring. And we will also prove that a C-M local ring is to be Gorenstein under some additional condition about the tight closure.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1855-1861
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• 2013

Let R be a commutative Noetherian ring, I an ideal of R, M and N two R-modules. We characterize the least integer i such that $H^i_I(M,N)$ is not weakly Artinian by using the notion of weakly filter regular sequences. Also, a local-global principle for minimax generalized local cohomology modules is shown and the result generalizes the corresponding result for local cohomology modules.

• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.633-643
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• 2010

Let a be an ideal of a commutative Noetherian ring R, M a finitely generated R-module and N a weakly Laskerian R-module. We show that if N has finite dimension d, then $Ass_R(H^d_a(N))$ consists of finitely many maximal ideals of R. Also, we find the least integer i, such that $H^i_a$(M, N) is not consisting of finitely many maximal ideals of R.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1341-1352
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• 2016

Let R be a commutative Noetherian ring, ${\Phi}$ a system of ideals of R and $I{\in}{\Phi}$. In this paper among other things we prove that if M is finitely generated and $t{\in}\mathbb{N}$ such that the R-module $H^i_{\Phi}(M)$ is $FD_{{\leq}1}$ (or weakly Laskerian) for all i < t, then $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all i < t and for any $FD_{{\leq}0}$ (or minimax) submodule N of $H^t_{\Phi}(M)$, the R-modules $Hom_R(R/I,H^t_{\Phi}(M)/N)$ and $Ext^1_R(R/I,H^t_{\Phi}(M)/N)$ are finitely generated. Also it is shown that if cd I = 1 or $dimM/IM{\leq}1$ (e.g., $dim\;R/I{\leq}1$) for all $I{\in}{\Phi}$, then the local cohomology module $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all $i{\geq}0$. These generalize the main results of Aghapournahr and Bahmanpour [2], Bahmanpour and Naghipour [6, 7]. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals.