• Title, Summary, Keyword: $\upsilon$-ring

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PRUFER ${\upsilon}$-MULTIPLICATION DOMAINS IN WHICH EACH t-IDEAL IS DIVISORIAL

  • Hwang, Chul-Ju;Chang, Gyu-Whan
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.259-268
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    • 1998
  • We give several characterizations of a TV-PVMD and we show that the localization R[X;S]$_{N_{\upsilon}}$ of a semigroup ring R[X;S] is a TV-PVMD if and only if R is a TV-PVMD where $N_{\upsilon}\;=\;\{f\;{\in}\;R[X]{\mid}(A_f)_{\upsilon} = R\}$ and S is a torsion free cancellative semigroup with zero.

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PIERCE STALKS OF EXCHANGE RINGS

  • Chen, Huanyin
    • Journal of the Korean Mathematical Society
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    • v.47 no.4
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    • pp.819-830
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    • 2010
  • We prove, in this article, that a ring R is a stable exchange ring if and only if so are all its Pierce stalks. If every Pierce stalks of R is artinian, then $1_R$ = u + $\upsilon$ with u, $\upsilon$ $\in$ U(R) if and only if for any a $\in$ R, there exist u, $\upsilon$ $\in$ U(R) such that a = u + $\upsilon$. Furthermore, there exists u $\in$ U(R) such that $1_R\;{\pm}\;u\;\in\;U(R)$ if and only if for any a $\in$ R, there exists u $\in$ U(R) such that $a\;{\pm}\;u\;\in\;U(R)$. We will give analogues to normal exchange rings. The root properties of such exchange rings are also obtained.

SIMPLE VALUATION IDEALS OF ORDER 3 IN TWO-DIMENSIONAL REGULAR LOCAL RINGS

  • Noh, Sun-Sook
    • 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 $b_{\upsilon}$.

GRADED INTEGRAL DOMAINS AND NAGATA RINGS, II

  • Chang, Gyu Whan
    • Korean Journal of Mathematics
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    • v.25 no.2
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    • pp.215-227
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    • 2017
  • Let D be an integral domain with quotient field K, X be an indeterminate over D, K[X] be the polynomial ring over K, and $R=\{f{\in}K[X]{\mid}f(0){\in}D\}$; so R is a subring of K[X] containing D[X]. For $f=a_0+a_1X+{\cdots}+a_nX^n{\in}R$, let C(f) be the ideal of R generated by $a_0$, $a_1X$, ${\ldots}$, $a_nX^n$ and $N(H)=\{g{\in}R{\mid}C(g)_{\upsilon}=R\}$. In this paper, we study two rings $R_{N(H)}$ and $Kr(R,{\upsilon})=\{{\frac{f}{g}}{\mid}f,g{\in}R,\;g{\neq}0,{\text{ and }}C(f){\subseteq}C(g)_{\upsilon}\}$. We then use these two rings to give some examples which show that the results of [4] are the best generalizations of Nagata rings and Kronecker function rings to graded integral domains.

Inhibition of Phospholipase $C{\Upsilon}1$ and Cancer Cell Proliferation by Lignans and Flavans from Machilus thunbergii

  • Lee, Ji-Suk;Kim, Jin-Woong;Yu, Young-Uck;Kim , Young-Choong
    • Archives of Pharmacal Research
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    • v.27 no.10
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    • pp.1043-1047
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    • 2004
  • Thirteen compounds were isolated from the $CH_2Cl_2$ fraction of Machilus thunbergii as phospholipase $C{\Upsilon}1\;(PLC{\Upsilon}1)$ inhibitors. These compounds were identified as nine lignans, two neolignans, and two flavans by spectroscopic analysis. Of these, 5,7-di-O-methyl-3',4'-methylenated (-)-epicatechin (12) and 5,7,3'-tri-O-methyl (-)-epicatechin (13) have not been reported previously in this plant. In addition, seven compounds, machilin A (1), (-)-sesamin (3), machilin G (5), (+)-galbacin (9), licarin A (10), (-)-acuminatin (11) and compound 12 showed dose-dependent potent inhibitory activities against $PLC{\Upsilon}1$ in vitro with $IC_{50}$ values ranging from 8.8 to 26.0 ${\mu}M$. These lignans, neolignans, and flavans are presented as a new class of $PLC{\Upsilon}1$ inhibitors. The brief study of the structure activity relationship of these compounds suggested that the benzene ring with the methylene dioxy group is responsible for the expression of inhibitory activities against $PLC{\Upsilon}1$. Moreover, it is suggested that inhibition of $PLC{\Upsilon}1$ may be an important mechanism for an antiproliferative effect on the human cancer cells. Therefore, these inhibitors may be utilized as cancer chemotherapeutic and chemopreventive agents.

OVERRINGS OF THE KRONECKER FUNCTION RING Kr(D, *) OF A PRUFER *-MULTIPLICATION DOMAIN D

  • Chang, Gyu-Whan
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.5
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    • pp.1013-1018
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    • 2009
  • Let * be an e.a.b. star operation on an integrally closed domain D, and let $K\gamma$(D, *) be the Kronecker function ring of D. We show that if D is a P*MD, then the mapping $D_{\alpha}{\mapsto}K{\gamma}(D_{\alpha},\;{\upsilon})$ is a bijection from the set {$D_{\alpha}$} of *-linked overrings of D into the set of overrings of $K{\gamma}(D,\;{\upsilon})$. This is a generalization of [5, Proposition 32.19] that if D is a Pr$\ddot{u}$fer domain, then the mapping $D_{\alpha}{\mapsto}K_{\gamma}(D_{\alpha},\;b)$ is a one-to-one mapping from the set {$D_{\alpha}$} of overrings of D onto the set of overrings of $K_{\gamma}$(D, b).

CHARACTERIZATIONS OF GRADED PRÜFER ⋆-MULTIPLICATION DOMAINS

  • Sahandi, Parviz
    • Korean Journal of Mathematics
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    • v.22 no.1
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    • pp.181-206
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    • 2014
  • Let $R={\bigoplus}_{\alpha{\in}\Gamma}R_{\alpha}$ be a graded integral domain graded by an arbitrary grading torsionless monoid ${\Gamma}$, and ⋆ be a semistar operation on R. In this paper we define and study the graded integral domain analogue of ⋆-Nagata and Kronecker function rings of R with respect to ⋆. We say that R is a graded Pr$\ddot{u}$fer ⋆-multiplication domain if each nonzero finitely generated homogeneous ideal of R is ⋆$_f$-invertible. Using ⋆-Nagata and Kronecker function rings, we give several different equivalent conditions for R to be a graded Pr$\ddot{u}$fer ⋆-multiplication domain. In particular we give new characterizations for a graded integral domain, to be a $P{\upsilon}MD$.

On Comaximal Graphs of Near-rings

Variation of Cone Crack Shape and Impact Damage According to Impact Velocity in Ceramic Materials (세라믹에서 충격속도에 따른 충격손상 및 콘크랙 형상의 변화)

  • Oh, Sang-Yeob;Shin, Hyung-Seop;Suh, Chang-Min
    • Proceedings of the KSME Conference
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    • pp.383-388
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    • 2001
  • Effects of particle property variation of cone crack shape according to impact velocity in silicon carbide materials were investigated. The damage induced by spherical impact having different material and size was different according to materials. The size of ring cracks induced on the surface of specimen increased with increase of impact velocity within elastic contact conditions. The impact of steel particle produced larger ring cracks than that of SiC particle. In case of high impact velocity, the impact of SiC particle produced radial cracks by the elastic-plastic deformation at impact regions. Also percussion cone was formed from the back surface of specimen when particle size become large and its impact velocity exceeded a critical value. Increasing impact velocity, zenithal angle of cone cracks in SiC material was linearly decreasing not effect of impact particle size. An empirical equation, $\theta=\theta_{st}-\upsilon_p(180-\theta_{st})(\rho_p/\rho_s)^{1/2}/415$, was obtained from the test data as a function of quasi-static zenithal angle of cone crack($\theta_{st}$), the density of impact particle(${\rho}_p$) and specimen(${\rho}_s$). Applying this equation to the another materials, the variation of zenithal angle of cone crack could be predicted from the particle impact velocity.

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