• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1001-1017
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• 2023

Let R be a commutative graded ring with nonzero identity and n a positive integer. Our principal aim in this paper is to introduce and study the notions of graded n-irreducible and strongly graded n-irreducible ideals which are generalizations of n-irreducible and strongly n-irreducible ideals to the context of graded rings, respectively. A proper graded ideal I of R is called graded n-irreducible (respectively, strongly graded n-irreducible) if for each graded ideals I₁, . . . , I_n+1 of R, I = I₁ ∩ · · · ∩ I_n+1 (respectively, I₁ ∩ · · · ∩ I_n+1 ⊆ I ) implies that there are n of the I_i 's whose intersection is I (respectively, whose intersection is in I). In order to give a graded study to this notions, we give the graded version of several other results, some of them are well known. Finally, as a special result, we give an example of a graded n-irreducible ideal which is not an n-irreducible ideal and an example of a graded ideal which is graded n-irreducible, but not graded (n - 1)-irreducible.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.373-386
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• 2014

Let (R,m) be a commutative Noetherian local ring. In this paper we show that a finitely generated R-module M of dimension d is Cohen-Macaulay if and only if there exists a proper ideal I of R such that depth($M/I^nM$) = d for $n{\gg}0$. Also we show that, if dim(R) = d and $I_1{\subset}\;{\cdots}\;{\subset}I_n$ is a chain of ideals of R such that $R/I_k$ is maximal Cohen-Macaulay for all k, then $n{\leq}{\ell}_R(R/(a_1,{\ldots},a_d)R)$ for every system of parameters $a1,{\ldots},a_d$ of R. Also, in the case where dim(R) = 2, we prove that the ideal transform $D_m(R/p)$ is minimax balanced big Cohen-Macaulay, for every $p{\in}Assh_R$(R), and we give some equivalent conditions for this ideal transform being maximal Cohen-Macaulay.

• East Asian mathematical journal
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• v.19 no.2
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• pp.233-240
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• 2003

Xie and Wu introduced an m-system in a po-semigroup. Kehayopulu gave characterizations of weakly prime ideals of po-semigroups and Lee and Kwon add two characterizations for weakly prime ideals. In this paper, we give a characterization of weakly prime ideals and a characterization of weakly semi-prime ideals in po-semigroups using m-system and n-system, respectively

• Communications of the Korean Mathematical Society
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• v.35 no.4
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• pp.1045-1056
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• 2020

Let R be a commutative ring with unity. The extension of annihilating-ideal graph of R, $^{\bar{\mathbb{AG}}}$(R), is the graph whose vertices are nonzero annihilating ideals of R and two distinct vertices I and J are adjacent if and only if there exist n, m ∈ ℕ such that IⁿJ^m = (0) with Iⁿ, J^m ≠ (0). First, we differentiate when 𝔸𝔾(R) and $^{\bar{\mathbb{AG}}}$(R) coincide. Then, we have characterized the diameter and the girth of $^{\bar{\mathbb{AG}}}$(R) when R is a finite direct products of rings. Moreover, we show that $^{\bar{\mathbb{AG}}}$(R) contains a cycle, if $^{\bar{\mathbb{AG}}}$(R) ≠ 𝔸𝔾(R).

• Bulletin of the Korean Mathematical Society
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• v.22 no.2
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• pp.117-119
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• 1985

Let R be a commutative noetherian ring with 1.neq.0, denoting by .nu.(I) the cardinality of a minimal basis of the ideal I. Let A be a polynomial ring in n>0 variables with coefficients in R, and let M be a maximal ideal of A. Generally it is shown that .nu.(M $A_{M}$).leq..nu.(M).leq..nu.(M $A_{M}$)+1. It is well known that the lower bound is not always satisfied, and the most classical examples occur in nonfactional Dedekind domains. But in many cases, (e.g., A is a polynomial ring whose coefficient ring is a field) the lower bound is attained. In [2] and [3], the conditions when the lower bound is satisfied is investigated. Especially in [3], it is shown that .nu.(M)=.nu.(M $A_{M}$) if M.cap.R=p is a maximal ideal or $A_{M}$ (equivalently $R_{p}$) is not regular or n>1. Hence the problem of determining whether .nu.(M)=.nu.(M $A_{M}$) can be studied when p is not maximal, $A_{M}$ is regular and n=1. The purpose of this note is to provide some conditions in which the lower bound is satisfied, when n=1 and R is a regular local ring (hence $A_{M}$ is regular)./ is regular).

• Korean Journal of Mathematics
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• v.4 no.1
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• pp.45-49
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• 1996

Let $F_0$ be the maximal real subfield of $\mathbb{Q}({\zeta}_q+{\zeta}_q^{-1})$ and $F_{\infty}={\cup}_{n{\geq}0}F_n$ be its basic $\mathbb{Z}_p$-extension. Let $A_n$ be the Sylow $p$-subgroup of the ideal class group of $F_n$. The aim of this paper is to examine the injectivity of the natural $mapA_n{\rightarrow}A_m$ induced by the inclusion $F_n{\rightarrow}F_m$ when $m>n{\geq}0$. By using cyclotomic units of $F_n$ and by applying cohomology theory, one gets the following result: If $p$ does not divide the order of $A_1$, then $A_n{\rightarrow}A_m$ is injective for all $m>n{\geq}0$.

• Kyungpook Mathematical Journal
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• v.58 no.1
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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.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.683-687
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• 1998

Suppose X is a subspace of $(\sum_{n=1} ^{\infty} X_n)_{c_0}$, dim $X_n<{\infty}$, which has the metric compact approximation property. It is proved that if Y is a Banach space of cotype q for some $2{\leq}1<{\infty}$ then K(X,Y) is an M-ideal in L(X,Y).

• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.887-893
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• 1996

Let (R,m) be a Cohen-Macaulay local ring with an infinite residue field and let $J = (a_1, \cdots, a_l)$ be a minimal reduction of an equimultiple ideal I of R. In this paper we shall prove that the following conditions are equivalent: (1) $I^2 = JI$. (2) $gr_I(R)/mgr_I(R)$ is Cohen-Macaulay with minimal multiplicity at its maximal homogeneous ideal N. (3) $N^2 = (a'_1, \cdots, a'_l)N$, where $a'_i$ denotes the images of $a_i$ in I/mI for $i = 1, \cdots, l$.

• Kyungpook Mathematical Journal
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• v.49 no.2
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• pp.283-288
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• 2009

Let N be a zero-symmetric near-ring with identity and let ${\Gamma}(N)$ be a graph with vertices as elements of N, where two different vertices a and b are adjacent if and only if + = N, where is the ideal of N generated by x. Let ${\Gamma}_1(N)$ be the subgraph of ${\Gamma}(N)$ generated by the set {n ${\in}$ N : = N} and ${\Gamma}_2(N)$ be the subgraph of ${\Gamma}(N)$ generated by the set $N{\backslash}{\upsilon}({\Gamma}_1(N))$, where ${\upsilon}(G)$ is the set of all vertices of a graph G. In this paper, we completely characterize the diameter of the subgraph ${\Gamma}_2(N)$ of ${\Gamma}(N)$. In addition, it is shown that for any near-ring, ${\Gamma}_2(N){\backslash}M(N)$ is a complete bipartite graph if and only if the number of maximal ideals of N is 2, where M(N) is the intersection of all maximal ideals of N and ${\Gamma}_2(N){\backslash}M(N)$ is the graph obtained by removing the elements of the set M(N) from the vertices set of the graph ${\Gamma}_2(N)$.