• 대한수학회보
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• 제49권5호
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• pp.1067-1079
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• 2012

Let M be a right R-module, (S, ${\leq}$) a strictly totally ordered monoid which is also artinian and ${\omega}:S{\rightarrow}Aut(R)$ a monoid homomorphism, and let $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ denote the generalized inverse polynomial module over the skew generalized power series ring [[$R^{S,{\leq}},{\omega}$]]. In this paper, we prove that $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same uniform dimension as its coefficient module $M_R$, and that if, in addition, R is a right perfect ring and S is a chain monoid, then $[M^{S,{\leq}}]_{[[R^{S,{\leq}},{\omega}]]$ has the same couniform dimension as its coefficient module $M_R$.

• Kyungpook Mathematical Journal
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• 제61권2호
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• pp.239-248
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• 2021

In this paper we study modules with the W F I⁺-extending property. We prove that if M satisfies the W F I⁺-extending, pseudo duo properties and M/(Soc M) has finite uniform dimension then M decompose into a direct sum of a semisimple submodule and a submodule of finite uniform dimension. In particular, if M satisfies the W F I⁺-extending, pseudo duo properties and ascending chain (respectively, descending chain) condition on essential submodules then M = M₁ ⊕ M₂ for some semisimple submodule M₁ and Noetherian (respectively, Artinian) submodule M₂. Moreover, we show that if M is a W F I-extending module with pseudo duo, C₂ and essential socle then the quotient ring of its endomorphism ring with Jacobson radical is a (von Neumann) regular ring. We provide several examples which illustrate our results.

• 대한수학회지
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• 제47권3호
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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.

• 호남수학학술지
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• 제38권2호
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• pp.295-304
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• 2016

In this paper we investigate the Artinianness of certain local cohomology modules $H^i_I(N)$ where N is a minimax module over a commutative Noetherian ring R and I is an ideal of R. Also, we characterize the set of attached prime ideals of $H^n_I(N)$, where n is the dimension of N.

• 대한수학회보
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• 제50권6호
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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.

• Kyungpook Mathematical Journal
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• 제58권1호
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• pp.9-17
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• 2018

Let $R ={\oplus}_{n{\in}Z}R_n$ be a graded commutative Noetherian ring and let I be a graded ideal of R and J be an arbitrary ideal. It is shown that the i-th generalized local cohomology module of graded module M with respect to the (I, J), is graded. Also, the asymptotic behaviour of the homogeneous components of $H^i_{I,J}(M)$ is investigated for some i's with a specified property.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.443-450
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• 2011

In this paper, we construct a Gorenstein Artinian algebra R/J with non-unimodal Hilbert function h = (1, 13, 12, 13, 1) to investigate the algebraic structure of the ideal J in a polynomial ring R. For this purpose, we use a software system Macaulay 2, which is devoted to supporting research in algebraic geometry and commutative algebra.

• Kyungpook Mathematical Journal
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• 제54권3호
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• pp.471-476
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• 2014

Let R be a ring. A right R-module M is PS-injective if every R-homomorphism $f:aR{\rightarrow}M$ for every principally small right ideal aR can be extended to $R{\rightarrow}M$. We investigate, in this paper, rings whose simple singular modules are PS-injective. New characterizations of semiprimitive rings and semisimple Artinian rings are given.

• 대한수학회보
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• 제57권1호
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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}}.

• 대한수학회보
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• 제59권4호
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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)}.