• 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.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.

• Journal of the Korean Mathematical Society
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• v.53 no.6
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• pp.1225-1236
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• 2016

Let R be a commutative ring with $1{\neq}0$ and n a positive integer. In this article, we introduce the n-Krull dimension of R, denoted $dim_n\;R$, which is the supremum of the lengths of chains of n-absorbing ideals of R. We study the n-Krull dimension in several classes of commutative rings. For example, the n-Krull dimension of an Artinian ring is finite for every positive integer n. In particular, if R is an Artinian ring with k maximal ideals and l(R) is the length of a composition series for R, then $dim_n\;R=l(R)-k$ for some positive integer n. It is proved that a Noetherian domain R is a Dedekind domain if and only if $dim_n\;R=n$ for every positive integer n if and only if $dim_2\;R=2$. It is shown that Krull's (Generalized) Principal Ideal Theorem does not hold in general when prime ideals are replaced by n-absorbing ideals for some n > 1.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.711-720
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• 2021

Let R denote a commutative noetherian ring, and let 𝐱 := x1, …, xd be an R-regular sequence. Suppose that 𝖆 denotes a monomial ideal with respect to 𝐱. The first purpose of this article is to show that 𝖆 is irreducible if and only if 𝖆 is a generalized-parametric ideal. Next, it is shown that, for any integer n ≥ 1, (x1, …, xd)n = ⋂P(f), where the intersection (irredundant) is taken over all monomials f = xe11 ⋯ xedd such that deg(f) = n - 1 and P(f) := (xe1+11, ⋯, xed+1d). The second main result of this paper shows that if 𝖖 := (𝐱) is a prime ideal of R which is contained in the Jacobson radical of R and R is 𝖖-adically complete, then 𝖆 is a parameter ideal if and only if 𝖆 is a monomial irreducible ideal and Rad(𝖆) = 𝖖. In addition, if a is generated by monomials m1, …, mr, then Rad(𝖆), the radical of a, is also monomial and Rad(𝖆) = (ω1, …, ωr), where ωi = rad(mi) for all i = 1, …, r.

• Bulletin of the Korean Mathematical Society
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• v.59 no.4
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• pp.917-928
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• 2022

Let R be a commutative Noetherian ring and I an ideal of R. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the finiteness of generalized local cohomology modules. Let M be a finitely generated R-module and N a representable R-module. We introduce the notions of the representation dimension r^I(M, N) and artinianness dimension a^I(M, N) of M, N with respect to I by r^I(M, N) = inf{i ∈ ℕ₀ : H^I_i(M, N) is not representable} and a^I(M, N) = inf{i ∈ ℕ₀ : H^I_i(M, N) is not artinian} and we show that a^I(M, N) = r^I(M, N) = inf{r^IR_𝔭 (M_𝔭,𝔭N) : 𝔭 ∈ Spec(R)} ≥ inf{a^IR_𝔭 (M_𝔭,𝔭N) : 𝔭 ∈ Spec(R)}. Also, in the case where R is semi-local and N a semi discrete linearly compact R-module such that N/∩_t>0I^tN is artinian we prove that inf{i : H^I_i(M, N) is not minimax}=inf{r^IR_𝔭 (M_𝔭,𝔭N) : 𝔭 ∈ Spec(R)＼Max(R)}.

• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.39-46
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• 2023

Let R be a commutative ring, M be a Noetherian R-module, and N a 2-absorbing submodule of M such that r(N :_R M) = 𝖕 is a prime ideal of R. The main result of the paper states that if N = Q₁ ∩ ⋯ ∩ Q_n with r(Q_i :_R M) = 𝖕_i, for i = 1, . . . , n, is a minimal primary decomposition of N, then the following statements are true. (i) 𝖕 = 𝖕_k for some 1 ≤ k ≤ n. (ii) For each j = 1, . . . , n there exists m_j ∈ M such that 𝖕_j = (N :_R m_j). (iii) For each i, j = 1, . . . , n either 𝖕_i ⊆ 𝖕_j or 𝖕_j ⊆ 𝖕_i. Let Γ_E(M) denote the zero-divisor graph of equivalence classes of zero divisors of M. It is shown that {Q₁∩ ⋯ ∩Q_n-1, Q₁∩ ⋯ ∩Q_n-2, . . . , Q₁} is an independent subset of V (Γ_E(M)), whenever the zero submodule of M is a 2-absorbing submodule and Q₁ ∩ ⋯ ∩ Q_n = 0 is its minimal primary decomposition. Furthermore, it is proved that Γ_E(M)[(0 :_R M)], the induced subgraph of Γ_E(M) by (0 :_R M), is complete.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1323-1336
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• 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.

• Journal of the Korean Mathematical Society
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• v.54 no.4
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• pp.1281-1299
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• 2017

A submodule N of a module M is called ${\mathcal{S}}$-closed (in M) if M/N is nonsingular. It is well-known that the class Closed of short exact sequences determined by closed submodules is a proper class in the sense of Buchsbaum. However, the class $\mathcal{S}-Closed$ of short exact sequences determined by $\mathcal{S}$-closed submodules need not be a proper class. In the first part of the paper, we describe the smallest proper class ${\langle}\mathcal{S-Closed}{\rangle}$ containing $\mathcal{S-Closed}$ in terms of $\mathcal{S}$-closed submodules. We show that this class coincides with the proper classes projectively generated by Goldie torsion modules and coprojectively generated by nonsingular modules. Moreover, for a right nonsingular ring R, it coincides with the proper class generated by neat submodules if and only if R is a right SI-ring. In abelian groups, the elements of this class are exactly torsionsplitting. In the second part, coprojective modules of this class which we call ec-flat modules are also investigated. We prove that injective modules are ec-flat if and only if each injective hull of a Goldie torsion module is projective if and only if every Goldie torsion module embeds in a projective module. For a left Noetherian right nonsingular ring R of which the identity element is a sum of orthogonal primitive idempotents, we prove that the class ${\langle}\mathcal{S-Closed}{\rangle}$ coincides with the class of pure-exact sequences of modules if and only if R is a two-sided hereditary, two-sided $\mathcal{CS}$-ring and every singular right module is a direct sum of finitely presented modules.

• Journal of the Korean Mathematical Society
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• v.55 no.6
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• pp.1381-1388
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• 2018

Let ${\Delta}$ be a simplicial complex, $I_{\Delta}$ its Stanley-Reisner ideal and $K[{\Delta}]$ its Stanley-Reisner ring over a field K. Assume that ${\Gamma}(R)$ denotes the zero-divisor graph of a commutative ring R. Here, first we present a condition on two reduced Noetherian rings R and R', equivalent to ${\Gamma}(R){\cong}{\Gamma}(R{^{\prime}})$. In particular, we show that ${\Gamma}(K[{\Delta}]){\cong}{\Gamma}(K^{\prime}[{\Delta}^{\prime}])$ if and only if ${\mid}Ass(I_{\Delta}){\mid}={\mid}Ass(I_{{{\Delta}^{\prime}}}){\mid}$ and either ${\mid}K{\mid}$, ${\mid}K^{\prime}{\mid}{\leq}{\aleph}_0$ or ${\mid}K{\mid}={\mid}K^{\prime}{\mid}$. This shows that ${\Gamma}(K[{\Delta}])$ contains little information about $K[{\Delta}]$. Then, we define the squarefree zero-divisor graph of $K[{\Delta}]$, denoted by ${\Gamma}_{sf}(K[{\Delta}])$, and prove that ${\Gamma}_{sf}(K[{\Delta}){\cong}{\Gamma}_{sf}(K[{\Delta}^{\prime}])$ if and only if $K[{\Delta}]{\cong}K[{\Delta}^{\prime}]$. Moreover, we show how to find dim $K[{\Delta}]$ and ${\mid}Ass(K[{\Delta}]){\mid}$ from ${\Gamma}_{sf}(K[{\Delta}])$.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.167-177
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• 2020

Let I be an ideal of Noetherian local ring (R, m) and M an artinian R-module. In this paper, we study colocalization of local homology modules. In fact we give Colocal-global Principle for the artinianness and minimaxness of local homology modules, which is a dual case of Local-global Principle for the finiteness of local cohomology modules. We define the representation dimension r^I (M) of M and the artinianness dimension a^I (M) of M relative to I by r^I (M) = inf{i ∈ ℕ₀ : H^I_i (M) is not representable}, and a^I (M) = inf{i ∈ ℕ₀ : H^I_i (M) is not artinian} and we will prove that i) a^I (M) = r^I (M) = inf{r^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R)} ≥ inf{a^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R)}, ii) inf{i ∈ ℕ₀ : H^I_i (M) is not minimax} = inf{r^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R) ∖ {𝔪}}. Also, we define the upper representation dimension R^I (M) of M relative to I by R^I (M) = sup{i ∈ ℕ₀ : H^I_i (M) is not representable}, and we will show that i) sup{i ∈ ℕ₀ : H^I_i (M) ≠ 0} = sup{i ∈ ℕ₀ : H^I_i (M) is not artinian} = sup{R^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R)}, ii) sup{i ∈ ℕ₀ : H^I_i (M) is not finitely generated} = sup{i ∈ ℕ₀ : H^I_i (M) is not minimax} = sup{R^IR_𝖕 (_𝖕M) : 𝖕 ∈ Spec(R) ∖ {𝔪}}.