• Journal of the Korean Mathematical Society
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• v.59 no.2
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• pp.379-405
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• 2022

We introduce the notions of Koszul N-complex, Čech N-complex and telescope N-complex, explicit derived torsion and derived completion functors in the derived category D_N (R) of N-complexes using the Čech N-complex and the telescope N-complex. Moreover, we give an equivalence between the categories of cohomologically 𝖆-torsion N-complexes and cohomologically 𝖆-adic complete N-complexes, and prove that over a commutative Noetherian ring, via Koszul cohomology, via RHom cohomology (resp. ⊗ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.149-160
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• 2023

Let 𝖆 and 𝖇 be ideals of a commutative Noetherian ring R and M a finitely generated R-module of finite dimension d > 0. In this paper, we obtain some results about the annihilators and attached primes of top local cohomology and top formal local cohomology modules. In particular, we determine Ann(𝖇 H^d_𝖆(M)), Att(𝖇 H^d_𝖆(M)), Ann(𝖇𝔉^d_𝖆(M)) and Att(𝖇𝔉^d_𝖆(M)).

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.71-77
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• 2024

Let A be a commutative integral domain with identity element and S a multiplicatively closed subset of A. In this paper, we introduce the concept of S-valuation domains as follows. The ring A is said to be an S-valuation domain if for every two ideals I and J of A, there exists s ∈ S such that either sI ⊆ J or sJ ⊆ I. We investigate some basic properties of S-valuation domains. Many examples and counterexamples are provided.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.379-395
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• 2018

Assume that R is a commutative ring with non-zero identity which is not an integral domain. An ideal I of R is called an annihilating ideal if there exists a non-zero element $a{\in}R$ such that Ia = 0. S. Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of R, denoted by ${\Omega}(R)$, as the graph with the vertex-set $A(R)^*$, the set of all non-zero annihilating ideals of R, and two distinct vertices I and J are adjacent if I + J is an annihilating ideal. In this paper, we study the relations between the diameters of ${\Omega}(R)$ and ${\Omega}(R[x])$. Also, we study the relations between the diameters of ${\Omega}(R)$ and ${\Omega}(R[[x]])$, whenever R is a Noetherian ring. In addition, we investigate the relations between the diameters of this graph and the zero-divisor graph. Moreover, we study some combinatorial properties of ${\Omega}(R)$ such as domination number and independence number. Furthermore, we study the complement of this graph.

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.437-442
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• 2005

In studying injective covers, the modules C such that Hom(E, C) = 0 and $Ext^1$(E, C) = 0 for all injective module E play an important role because of Wakamatsu's lemma. If C is a module over the ring k[[x, y]] with k a field, the class of these modules C contains the class $\={D}$ of all direct summands of products of modules of finite length ([3, Theorem 2.9]). In this paper we show that every module over any commutative ring has a $\={D}$-preenvelope.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.715-736
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• 2012

Brown provided a structure theorem for a class of perfect ideals of grade 3 with type ${\lambda}$ > 0. We introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4 in a Noetherian local ring. We construct a class of perfect ideals I of grade 3 with type 2 defined by a certain skew-symmetrizable matrix. We present the Hilbert function of the standard $k$-algebras R/I, where R is the polynomial ring $R=k[v_0,v_1,{\ldots},v_m]$ over a field $k$ with indeterminates $v_i$ and deg $v_i=1$.

• Communications of the Korean Mathematical Society
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• v.20 no.2
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• pp.275-290
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• 2005

Let $S\;=\;R[\chi_{ij}\mid1\;{\le}\;i\;{\le}\;m,\;1\;{\le}\;j\;{\le}\;n]$ be the polynomial ring over a noetherian commutative ring R and $I_p$ be the determinantal ideal generated by the $p\;\times\;p$ minors of the generic matrix $(\chi_{ij})(1{\le}P{\le}min(m,n))$. We describe a minimal free resolution of $S/I_{p}$, in the case m = n = p + 2 over $\mathbb{Z}$.

• Kyungpook Mathematical Journal
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• v.61 no.1
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• pp.23-32
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• 2021

Let D be an integral domain with quotient field K, X an indeterminate over D, ∗ a star operation on D, and Cl∗ (D) be the ∗-class group of D. The ∗w-operation on D is a star operation defined by I∗w = {x ∈ K | xJ ⊆ I for a nonzero finitely generated ideal J of D with J∗ = D}. In this paper, we study two star operations {∗} and [∗] on D[X] defined by A{∗} = ∩P∈∗w-Max(D) ADP [X] and A[∗] = (∩P∈∗w-Max(D) AD[X]P[X]) ∩ AK[X]. Among other things, we show that Cl∗(D) ≅ Cl[∗](D[X]) if and only if D is integrally closed.

• 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.

• Communications of the Korean Mathematical Society
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• v.38 no.3
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• pp.663-677
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• 2023

The purpose of this paper is to introduce a new class of rings containing the class of SFT-rings and contained in the class of rings with Noetherian prime spectrum. Let A be a commutative ring with unit and I be an ideal of A. We say that I is SFT if there exist an integer k ≥ 1 and a finitely generated ideal F ⊆ I of A such that x^k ∈ F for every x ∈ I. The ring A is said to be nonnil-SFT, if each nonnil-ideal (i.e., not contained in the nilradical of A) is SFT. We investigate the nonnil-SFT variant of some well known theorems on SFT-rings. Also we study the transfer of this property to Nagata's idealization and the amalgamation algebra along an ideal. Many examples are given. In fact, using the amalgamation construction, we give an infinite family of nonnil-SFT rings which are not SFT.