• Communications of the Korean Mathematical Society
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• v.13 no.1
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• pp.1-6
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• 1998

The paper refers to ordered semigroups in which $x^2 (x \in S)$ are left ideal elements. We mainly show that this $po$-semigroup is left regular if and only if S is a union of left simple subsemigroups of S.

• 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).

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.511-528
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• 2008

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and $\upsilon$ be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple $\upsilon$-ideals $m\;=\;P_0\;{\supset}\;P_1\;{\supset}\;{\cdots}\;{\supset}\;P_t\;=\;P$ and all the other $\upsilon$-ideals are uniquely factored into a product of those simple ones [17]. Lipman further showed that the predecessor of the smallest simple $\upsilon$-ideal P is either simple or the product of two simple $\upsilon$-ideals. The simple integrally closed ideal P is said to be free for the former and satellite for the later. In this paper we describe the sequence of simple $\upsilon$-ideals when P is satellite of order 3 in terms of the invariant $b_{\upsilon}\;=\;|\upsilon(x)\;-\;\upsilon(y)|$, where $\upsilon$ is the prime divisor associated to P and m = (x, y). Denote $b_{\upsilon}$ by b and let b = 3k + 1 for k = 0, 1, 2. Let $n_i$ be the number of nonmaximal simple $\upsilon$-ideals of order i for i = 1, 2, 3. We show that the numbers $n_{\upsilon}$ = ($n_1$, $n_2$, $n_3$) = (${\lceil}\frac{b+1}{3}{\rceil}$, 1, 1) and that the rank of P is ${\lceil}\frac{b+7}{3}{\rceil}$ = k + 3. We then describe all the $\upsilon$-ideals from m to P as products of those simple $\upsilon$-ideals. In particular, we find the conductor ideal and the $\upsilon$-predecessor of the given ideal P in cases of b = 1, 2 and for b = 3k + 1, 3k + 2, 3k for $k\;{\geq}\;1$. We also find the value semigroup $\upsilon(R)$ of a satellite simple valuation ideal P of order 3 in terms of $b_{\upsilon}$.

• Kyungpook Mathematical Journal
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• v.47 no.4
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• pp.473-480
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• 2007

In this paper we define prime right ideals of semirings and prove that if every right ideal of a semiring R is prime then R is weakly regular. We also prove that if the set of right ideals of R is totally ordered then every right ideal of R is prime if and only if R is right weakly regular. Moreover in this paper we also define prime subsemimodule (generalizing the concept of prime right ideals) of an R-semimodule. We prove that if a subsemimodule K of an R-semimodule M is prime then $A_K(M)$ is also a prime ideal of R.

• Communications of the Korean Mathematical Society
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• v.39 no.1
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• pp.59-69
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• 2024

Let R be a commutative ring. An R-module E is said to be regular injective provided that Ext¹_R(R/I, E) = 0 for any regular ideal I of R. We first show that the class of regular injective modules have the hereditary property, and then introduce and study the regular injective dimension of modules and regular global dimension of rings. Finally, we give some homological characterizations of total rings of quotients and Dedekind rings.

• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.1-15
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• 2010

We introduce, in this article, the quasi-stable exchange ideal for associative rings. If I is a quasi-stable exchange ideal of a ring R, then so is $M_n$(I) as an ideal of $M_n$(R). As an application, we prove that every square regular matrix over quasi-stable exchange ideal admits a diagonal reduction by quasi invertible matrices. Examples of such ideals are given as well.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.213-226
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• 2022

Let R be a ring and M an R-module. Then M is said to be regular w-flat provided that the natural homomorphism I ⊗_R M → R ⊗_R M is a w-monomorphism for any regular ideal I. We distinguish regular w-flat modules from regular flat modules and w-flat modules by idealization constructions. Then we give some characterizations of total quotient rings and Prüfer v-multiplication rings (PvMRs for short) utilizing the homological properties of regular w-flat modules.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.799-809
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• 2013

An ideal I of a local ring (R, m, k) is said to be m-full if there exists an element $x{\in}m$ such that Im : x = I. An ideal I of a local ring R is said to have the Rees property if ${\mu}$(I) > ${\mu}$(J) for any ideal J containing I. We study properties of m-full ideals and we characterize m-primary m-full ideals in terms of the minimal number of generators of the ideals. In particular, for a m-primary ideal I of a 2-dimensional regular local ring (R, m, k), we will show that the following conditions are equivalent. 1. I is m-full 2. I has the Rees property 3. ${\mu}$(I)=o(I)+1 In this paper, let (R, m, k) be a commutative Noetherian local ring with infinite residue field k = R/m.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.99-124
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• 2017

The structure theorems [3, 6, 21] for the classes of perfect ideals of grade 3 have been generalized to the structure theorems for the classes of perfect ideals linked to almost complete intersections of grade 3 by a regular sequence [15]. In this paper we obtain structure theorems for two classes of Gorenstein ideals of grade 4 expressed as the sum of a perfect ideal of grade 3 (except a Gorenstein ideal of grade 3) and an almost complete intersection of grade 3 which are geometrically linked by a regular sequence.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.12 no.3
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• pp.215-225
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• 2012

We initiate the study of interval-valued fuzzy quasi-ideal of a semigroup. In Section 2, we list some basic definitions in the later sections. In Section 3, we investigate interval-valued fuzzy subsemigroups and in Section 4, we define interval-valued fuzzy quasi-ideals and establish some of their basic properties. In Section 5, we obtain characterizations of regular and intraregular semigroups using the machinery developed in the preceding sections.