• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.871-878
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• 2007

We define a numerical invariant $row_{CM}(A)$ over Cohen-Macaulay local ring A, which is related to rows of the presenting matrices of maximal Cohen-Macaulay modules without free summands. We show that $row(A)=row_{CM}(A)$ for a Cohen-Macaulay(not necessarily Gorenstein) local ring A.

• Communications of the Korean Mathematical Society
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• v.11 no.3
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• pp.569-573
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• 1996

We study some special cases of Chow groups of a ramified complete regular local ring R of dimension n. We prove that (a) for codimension 3 Gorenstein ideal I, [I] = 0 in $A_{n-3}(R)$ and (b) for a particular class of almost complete intersection prime ideals P of height i, [P] = 0 in $A_{n-i}(R)$.

• Journal of the Korean Mathematical Society
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• v.54 no.5
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• pp.1379-1410
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• 2017

Kang and Ko introduced a skew-symmetrizable matrix to describe a structure theorem for complete intersections of grade 4. Let $R=k[w_0,\;w_1,\;w_2,\;{\ldots},\;w_m]$ be the polynomial ring over an algebraically closed field k with indetermiantes $w_l$ and deg $w_l=1$, and $I_i$ a homogeneous perfect ideal of grade 3 with type $t_i$ defined by a skew-symmetrizable matrix $G_i(1{\leq}t_i{\leq}4)$. We show that for m = 2 the Hilbert function of the zero dimensional standard k-algebra $R/I_i$ is determined by CI-sequences and a Gorenstein sequence. As an application of this result we show that for i = 1, 2, 3 and for m = 3 a Gorenstein sequence $h(R/H_i)=(1,\;4,\;h_2,\;{\ldots},\;h_s)$ is unimodal, where $H_i$ is the sum of homogeneous perfect ideals $I_i$ and $J_i$ which are geometrically linked by a homogeneous regular sequence z in $I_i{\cap}J_i$.

• Bulletin of the Korean Mathematical Society
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• v.54 no.4
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• pp.1331-1336
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• 2017

Assume that R is a commutative Noetherian ring with non-zero identity, a is an ideal of R, X is an R-module, and t is a non-negative integer. In this paper, we present upper bounds for the injective dimension of X in terms of the injective dimensions of its local cohomology modules and an upper bound for the injective dimension of $H^t_{\alpha}(X)$ in terms of the injective dimensions of the modules $H^i_{\alpha}(X)$, $i{\neq}t$, and that of X. As a consequence, we observe that R is Gorenstein whenever $H^t_{\alpha}(R)$ is of finite injective dimension for all i.

• Journal of Integrative Natural Science
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• v.4 no.4
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• pp.278-281
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• 2011

We discuss an application of 'restrictions on the entries of the maps in the minimal free resolution' and '$SC_r$-condition of modules', and give an alternative proof of the following result of Foxby: Let M be a finitely generated module of dimension over a Noetherian local ring (A,m). Suppose that $\hat{A}$ has no embedded primes. If A is not Gorenstein, then ${\mu}_i(m,A){\geq}2$ for all i ${\geq}$ dimA.

• Bulletin of the Korean Mathematical Society
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• v.58 no.6
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• pp.1387-1400
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• 2021

Suppose that $T=\(\array{A&0\\U&B}\)$ is a formal triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. Let ℭ₁ and ℭ₂ be two classes of left A-modules, 𝔇₁ and 𝔇₂ be two classes of left B-modules. We prove that (ℭ₁, ℭ₂) and (𝔇₁, 𝔇₂) are admissible balanced pairs if and only if (p(ℭ₁, 𝔇₁), h(ℭ₂, 𝔇₂) is an admissible balanced pair in T-Mod. Furthermore, we describe when ($P^{C_1}_{D_1}$, $I^{C_2}_{D_2}$) is an admissible balanced pair in T-Mod. As a consequence, we characterize when T is a left virtually Gorenstein ring.

• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.137-147
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• 2015

We derive in the paper the tensor product functor -${\otimes}_R$- by using proper $\mathcal{GP}_C$-resolutions, where C is a semidualizing module. After giving several cases in which different relative homologies agree, we use the Pontryagin duals of $\mathcal{G}_C$-projective modules to establish a balance result for such relative homology over a Cohen-Macaulay ring with a dualizing module D.

• Journal of the Korean Mathematical Society
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• v.58 no.6
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• pp.1367-1383
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• 2021

This paper studies Young diagrams of symmetric and pseudo-symmetric numerical semigroups and describes new operations on Young diagrams as well as numerical semigroups. These provide new decompositions of symmetric and pseudo-symmetric semigroups into a numerical semigroup and its dual. It is also given exactly for what kind of numerical semigroup S, the semigroup ring 𝕜⟦S⟧ has at least one Gorenstein subring and has at least one Kunz subring.

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.29-40
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• 2024

Let S and R be rings and _SC_R a semidualizing bimodule. We introduce the notion of G_C-FP_n-injective modules, which generalizes G_C-FP-injective modules and G_C-weak injective modules. The homological properties and the stability of G_C-FP_n-injective modules are investigated. When S is a left n-coherent ring, several nice properties and new Foxby equivalences relative to G_C-FP_n-injective modules are given.

• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.417-429
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• 2015

Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by ${\Gamma}_{Ann}(R)$, is a graph whose vertices are all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if $I{\cap}Ann(J){\neq}\{0\}$ or $J{\cap}Ann(I){\neq}\{0\}$. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose annihilator ideal graphs are totally disconnected. Also, we study diameter, girth, clique number and chromatic number of this graph. Moreover, we study some relations between annihilator ideal graph and zero-divisor graph associated with R. Among other results, it is proved that for a Noetherian ring R if ${\Gamma}_{Ann}(R)$ is triangle free, then R is Gorenstein.