• 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 $\mathbb{Z}$.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.385-402
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• 2015

In this paper we study the (good) semi-prime closure operations on ideals of a BCK-algebra, lower BCK-semilattice, Noetherian BCK-algebra and meet quotient ideal and then we give several theorems that make different (good) semi-prime closure operations. Moreover by given some examples we show that the given different notions are independent together, for instance there is a semi-prime closure operation, which is not a good semi-prime. Finally by given the notion of "$c_f$-Max X", we prove that every member of "$c_f$-Max X" is a prime ideal. Also we conclude some more related results.

• Communications of the Korean Mathematical Society
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• v.31 no.4
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• pp.695-701
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• 2016

Let I and J be two ideals of a commutative Noetherian ring R and M, N be two non-zero finitely generated R-modules. Let t be a non-negative integer such that $H^i_{I,J}(N)$ is (I, J)-minimax for all i < t. It is shown that the generalized local cohomology module $H^i_{I,J}(M,N)$ is (I, J)-Cofinite minimax for all i < t. Also, we prove that the R-module $Ext^j_R(R/I,H^i_{I,J}(N))$ is finitely generated for all $i{\leq}t$ and j = 0, 1.

• Bulletin of the Korean Mathematical Society
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• v.56 no.2
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• pp.471-477
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• 2019

Let (R, m) be a commutative Noetherian ring, I an ideal of R and M a non-zero finitely generated R-module. We show that if M and $H_0(I,M)$ are aCM R-modules and $I=(x_1,{\cdots},x_{n+1})$ such that $x_1,{\cdots},x_n$ is an M-regular sequence, then $H_i(I,M)$ is an aCM R-module for all i. Moreover, we prove that if R and $H_i(I,R)$ are aCM for all i, then R/(0 : I) is aCM. In addition, we prove that if R is aCM and $x_1,{\cdots},x_n$ is an aCM d-sequence, then depth $H_i(x_1,{\cdots},x_n;R){\geq}i-1$ for all i.

• Communications of the Korean Mathematical Society
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• v.36 no.1
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• pp.1-10
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• 2021

Let R and S be two commutative rings, J be an ideal of S and f : R → S be a ring homomorphism. The amalgamation of R and S along J with respect to f, denoted by R ⋈f J, is the special subring of R × S defined by R ⋈f J = {(a, f(a) + j) | a ∈ R, j ∈ J}. In this paper, we study some basic properties of a special kind of R ⋈f J-modules, called the amalgamation of M and N along J with respect to ��, and defined by M ⋈�� JN := {(m, ��(m) + n) | m ∈ M and n ∈ JN}, where �� : M → N is an R-module homomorphism. The new results generalize some known results on the amalgamation of rings and the duplication of a module along an ideal.

• Kyungpook Mathematical Journal
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• v.61 no.2
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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.

• 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.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.407-423
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• 2022

In this paper, we determine the structure of rings that have secondary representation (called representable rings) and give some characterizations of these rings. Also, we characterize Artinian rings in terms of representable rings. Then we introduce completely representable modules, modules every non-zero submodule of which have secondary representation, and give some necessary and sufficient conditions for a module to be completely representable. Finally, we define strongly representable modules and give some conditions under which representable module is strongly representable.

• 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).

• 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.