• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.737-743
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• 2016

Let a and b be two ideals of a commutative Noetherian ring R, M a finitely generated R-module and i an integer. In this paper we study formal local cohomology modules with respect to a pair of ideals. We denote the i-th a-formal local cohomology module M with respect to b by ${\mathfrak{F}}^i_{a,b}(M)$. We show that if ${\mathfrak{F}}^i_{a,b}(M)$ is artinian, then $a{\subseteq}{\sqrt{(0:{\mathfrak{F}}^i_{a,b}(M))$. Also, we show that ${\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$ is artinian and we determine the set $Att_R\;{\mathfrak{F}}^{\text{dim }M}_{a,b}(M)$.

• 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.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1389-1421
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• 2018

In this paper, we introduce the notion of C-Gorenstein weak injective modules with respect to a semidualizing bimodule $_SC_R$, where R and S are arbitrary associative rings. We show that an iteration of the procedure used to define $G_C$-weak injective modules yields exactly the $G_C$-weak injective modules, and then give the Foxby equivalence in this setting analogous to that of C-Gorenstein injective modules over commutative Noetherian rings. Finally, some applications are given, including weak co-Auslander-Buchweitz context, model structure and dual pair induced by $G_C$-weak injective modules.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1385-1394
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• 2016

Let (R, m) be a Noetherian local ring, I, J two ideals of R, and A an Artinian R-module. Let $k{\geq}0$ be an integer and $r=Width_{>k}(I,A)$ the supremum of lengths of A-cosequences in dimension > k in I defined by Nhan-Hoang [9]. It is first shown that for each $t{\leq}r$ and each sequence $x_1,{\cdots},x_t$ which is an A-cosequence in dimension > k, the set $$\Large(\bigcup^{t}_{i=0}Att_R(0:_A(x_1^{n_1},{\ldots},x_i^{n_i})))_{{\geq}k}$$ is independent of the choice of $n_1,{\ldots},n_t$. Let r be the eventual value of $Width_{>k}(0:_AJ^n)$. Then our second result says that for each $t{\leq}r$ the set $\large(\bigcup\limits_{i=0}^{t}Att_R(Tor_i^R(R/I,\;(0:_AJ^n))))_{{\geq}k}$ is stable for large n.

• Bulletin of the Korean Mathematical Society
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• v.49 no.5
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• pp.989-996
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• 2012

Let R be a commutative Noetherian ring and let I be an ideal of R. In this paper we study the amalgamated duplication ring $R{\bowtie}I$ which is introduced by D'Anna and Fontana. It is shown that if R is generically Cohen-Macaulay (resp. generically Gorenstein) and I is generically maximal Cohen-Macaulay (resp. generically canonical module), then $R{\bowtie}I$ is generically Cohen-Macaulay (resp. generically Gorenstein). We also de ned generically quasi-Gorenstein ring and we investigate when $R{\bowtie}I$ is generically quasi-Gorenstein. In addition, it is shown that $R{\bowtie}I$ is approximately Cohen-Macaulay if and only if R is approximately Cohen-Macaulay, provided some special conditions. Finally it is shown that if R is approximately Gorenstein, then $R{\bowtie}I$ is approximately Gorenstein.

• 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.

• Journal of the Korean Mathematical Society
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• v.60 no.3
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• pp.683-694
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• 2023

Let a be an ideal of 𝔞 commutative noetherian ring R. We give some descriptions of the 𝔞-depth of 𝔞-relative Cohen-Macaulay modules by cohomological dimensions, and study how relative Cohen-Macaulayness behaves under flat extensions. As applications, the perseverance of relative Cohen-Macaulayness in a polynomial ring, formal power series ring and completion are given.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1125-1128
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• 2011

Let R be a commutative Noetherian ring, a an ideal of R, and M a minimax R-module. We prove that the local cohomology modules $H^j_a(M)$ are a-cominimax; that is, $Ext^i_R$(R/a, $H^j_a(M)$) is minimax for all i and j in the following cases: (a) dim R/a = 1; (b) cd(a) = 1, where cd is the cohomological dimension of a in R; (c) dim $R{\leq}2$. In these cases we also prove that the Bass numbers and the Betti numbers of $H^j_a(M)$ are finite.

• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.37-43
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• 2023

Let 𝔞 and 𝔟 be ideals of a commutative Noetherian ring R and M be a finitely generated R-module of dimension d > 0. We prove some results concerning the top local cohomology and top formal local cohomology modules. Among other things, we determine Supp_R(𝔟 H^d_𝔞(M)) and Supp_R(𝔟𝔉^d_𝔞(M)). Also, we obtain some relations between Ann_R(𝔟 H^d_𝔞(M)), Att_R(𝔟 H^d_𝔞(M)) and Supp_R(𝔟 H^d_𝔞(M)) and we get similar results for 𝔟𝔉^d_𝔞(M).

• The Pure and Applied Mathematics
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• v.1 no.1
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• pp.1-6
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• 1994

In this paper, some properties for a PI-ring satisfying the descending chain condition on essential left ideals are studied: Let R be a ring with a polynomial identity satisfying the descending chain condition on essential ideals. Then all minimal prime ideals in R are maximal ideals. Moreover, if R has only finitely many minimal prime ideals, then R is left and right Artinian. Consequently, if every primeideal of R is finitely generated as a left ideal, then R is left and right Artinian. A finitely generated PI-algebra over a commutative Noetherian ring satisfying the descending chain condition on essential left ideals is a finite module over its center.(omitted)