• 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.

• 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 ${\Gamma}_{sf}(K[{\Delta}])$.

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.1083-1096
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• 2012

For a given ideal I of a Noetherian ring R and an arbitrary integer ${\kappa}{\geq}-1$, we apply the concept of ${\kappa}$-regular sequences and the notion of ${\kappa}$-depth to give some results on modules called ${\kappa}$-Cohen Macaulay modules, which in local case, is exactly the ${\kappa}$-modules (as a generalization of f-modules). Meanwhile, we give an expression of local cohomology with respect to any ${\kappa}$-regular sequence in I, in a particular case. We prove that the dimension of homology modules of the Koszul complex with respect to any ${\kappa}$-regular sequence is at most ${\kappa}$. Therefore homology modules of the Koszul complex with respect to any filter regular sequence has finite length.

• Honam Mathematical Journal
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• v.39 no.2
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• pp.275-296
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• 2017

All rings considered here are commutative rings with identity and all modules considered here are unital left modules. A submodule N of an R-module M is said to be extended to M if $N=aM$ for some ideal a of R and it is said to be fully invariant if ${\varphi}(L){\subseteq}L$ for every ${\varphi}{\in}End(M)$. An R-module M is called a [resp., fully invariant] multiplication module if every [resp., fully invariant] submodule is extended to M. The class of fully invariant multiplication modules is bigger than the class of multiplication modules. We deal with prime submodules and associated prime submodules of fully invariant multiplication modules. In particular, when M is a nonzero faithful multiplication module over a Noetherian ring, we characterize the zero-divisors of M in terms of the associated prime submodules, and we show that the set Aps(M) of associated prime submodules of M determines the set $Zdv_M(M)$ of zero-dvisors of M and the support Supp(M) of M.

• 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}}.