• 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.

• Honam Mathematical Journal
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• v.21 no.1
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• pp.19-33
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• 1999

In this paper we reproduce results on $Gr{\ddot{o}}bner$ fans of ideals following a paper by T. Mora and L. Robbiano [5], and introduce stable $Gr{\ddot{o}}bner$ fans of ideals introduced by D. Mall [4]. We make minor corrections for the clarification, simplify proofs and provide new proofs. At the end we give a description of $Gr{\ddot{o}}bner$ fans of toric ideals.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.113-125
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• 2019

In this paper, we introduce the elimination ideal $I_D(G)$ associated to a simple finite graph G. We obtain the upper bound of Castelnuovo-Mumford regularity of elimination ideal for various classes of graphs.

• Communications of the Korean Mathematical Society
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• v.11 no.4
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• pp.895-902
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• 1996

In this paper, we describe the ideal generated by non-empty stable set in a BCI-group as a simple form, and obtain an equivalent condition of prime Z-ideal.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.615-624
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• 1998

We consider a generalization of [1. theorem 2.5]. We give a description of the element of $_{\mathcal I}^l$(resp. $_{\mathcal I} ^r$), where A and B are left (resp. right)${\mathcal I}$-ideals of an IS-algebra X. For a nonempty left (resp. right) stable, we obtain a condition for $_{\mathcal I}^l$(resp. $_{\mathcal I}^l$) to be closed. We give a characterization of a closed $\mathcal I$-ideal in an IS-algebra, and show that in a finite IS-algebra, every $\mathcal I$-ideal is closed.

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.793-807
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• 2020

Stable range of rings is a unifying concept for problems related to the substitution and cancellation of modules. The newly appeared element-wise setting for the simplest case of stable range one is tempting to study the lifting property modulo ideals. We study the lifting of elements having (idempotent) stable range one from a quotient of a ring R modulo a two-sided ideal I by providing several examples and investigating the relations with other lifting properties, including lifting idempotents, lifting units, and lifting of von Neumann regular elements. In the case where the ring R is a left or a right duo ring, we show that stable range one elements lift modulo every two-sided ideal if and only if R is a ring with stable range one. Under a mild assumption, we further prove that the lifting of elements having idempotent stable range one implies the lifting of von Neumann regular elements.

• Bulletin of the Korean Mathematical Society
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• v.53 no.5
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• pp.1385-1394
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• 2016

Let (R, m) be a Noetherian local ring, I, J two ideals of R, and A an Artinian R-module. Let $k{\geq}0$ be an integer and $r=Width_{>k}(I,A)$ the supremum of lengths of A-cosequences in dimension > k in I defined by Nhan-Hoang [9]. It is first shown that for each $t{\leq}r$ and each sequence $x_1,{\cdots},x_t$ which is an A-cosequence in dimension > k, the set $$\Large(\bigcup^{t}_{i=0}Att_R(0:_A(x_1^{n_1},{\ldots},x_i^{n_i})))_{{\geq}k}$$ is independent of the choice of $n_1,{\ldots},n_t$. Let r be the eventual value of $Width_{>k}(0:_AJ^n)$. Then our second result says that for each $t{\leq}r$ the set $\large(\bigcup\limits_{i=0}^{t}Att_R(Tor_i^R(R/I,\;(0:_AJ^n))))_{{\geq}k}$ is stable for large n.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.83-92
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• 2024

Let R be a commutative ring with nonzero identity and M be an R-module. In this paper, we first introduce the concept of S-idempotent element of R. Then we give a relation between S-idempotents of R and clopen sets of S-Zariski topology. After that we define S-pure ideal which is a generalization of the notion of pure ideal. In fact, every pure ideal is S-pure but the converse may not be true. Afterwards, we show that there is a relation between S-pure ideals of R and closed sets of S-Zariski topology that are stable under generalization.

• Cross-Cultural Studies
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• v.50
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• pp.483-507
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• 2018

Shylock in Shakespeare's play, The Merchant of Venice has been considered as either a devilish villain, or as a victim who was persecuted unfairly by the Christian society in Venice. By focusing on the matter of the Other, which has been summarily overlooked in literary texts and the literary criticism, it is noted that the New Historical and Cultural criticism interpreted Shylock as the racial, religious, and economic Other in the Venetian society which at the time was dominated by Christian ideals. The purpose of this paper is to show how Shylock becomes an abjected Other, that is, the abject, based on Julia Kristeva's theory of abjection. According to Kristeva, an abjection is the process of expulsion of otherness from society, through which the subject or the nation tries to set up clear boundaries and establish a stable identity. Shylock is marginalized and abjected by the borders drawn by the Venetian Christian society, which in a strong sense tries to protect its identity and homogeneity by rejecting and excluding any unclean or improper otherness. The borders include the two visible borders like the Ghetto and the red hats worn by the Jews, and one invisible border in the religious and economic fields. By asking for one pound of Antonio's flesh when he can't pay back 3,000 ducats owed, Shylock tries to cross the border between Christians and Jews. Portia frustrates Shylock's desire to violate the border by presenting a different interpretation of the expression, 'one pound of flesh,' from Shylock's interpretation. And in doing so she expels him back to his original position of abject.