• Journal for History of Mathematics
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• v.17 no.4
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• pp.37-44
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• 2004

Homological algebra has emerged and developed since 1950s. However, in 1890's Hilbert investigated the resolutions in his Syzygy Theorem which is a vital ingredient in homological algebra. In 1956 Serre has proved the finite global dimension of regular local rings. His result give a basic tool in homological algebra. This paper also deals with the depth formula that was raised by Auslander in 1961.

• Proceedings of the PSK Conference
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• 2003.04a
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• pp.250.2-250.2
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• 2003

A series of rutaecarpine homologues were prepared from 2,3-polymethylene-4(3H-quinazolinones in 4 steps [i) PhCHO/Ac$_2$O, ⅱ) O,$_3$ ⅲ) PhNHNH$_2$HCl, and ⅳ) PPA], in which dihedral angles of the two planar aromatic rings (indole and 4(3H)-quinazolinone) were controlled in a regular fashion. Their inhibitory activities on COX-1 and COX-2 were evaluated to show that the inhibitory activities were increased with the increase of the length of methylene unit while selectivities on COX-2 decreased leading a loss in trimethylene bridged system.

• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.121-122
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• 2023

In this erratum, we correct a mistake in the proof of Proposition 2.7. In fact the equivalence (3) ⇐ (4) "R is a quasi-regular ring if and only if R is a reduced ring and every principal ideal contained in Z(R) is a 0-ideal" does not hold as we only have Rx ⊆ O(S).

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.427-436
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• 2005

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 v 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 v-ideals $m=P_0\;{\supset}\;P_1\;{\supset}\;{\cdotS}\;{\supset}\;P_t=P$ and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by $n_v$. In this paper, if the m-adic order of P is 2, we show that $O(P_i)\;=\;1\;for\;1\;{\leq}\;i\; {\leq}\;{\lceil}\;{\frac{b+1}{2}}{\rceil}\;and\;O(P_i)\;=2\;for\;{\lceil}\;\frac{b+3}{2}\rceil\;{\leq}\;i\;\leq\;t,\;where\;b=n_v$. We also show that $n_w\;=\;n_v$ when w is the prime divisor associated to a simple v-ideal $Q\;{\supset}\;P$ of order 2 and that w(R) = v(R) as well.

• Journal of KIISE:Computer Systems and Theory
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• v.33 no.1_2
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• pp.105-109
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• 2006

Hypercube has a regular and hierarchical structure, therefore it can be applied to the development of efficient diagnosis algorithm. Kranakis and Pelc [7] have proposed HYP-DIAG algorithm to implement different method of HADA/IHADA and adaptive cube partition method after embedding the small size of ring that includes all the faulty nodes. In this paper, we propose new method to reduce testing rounds by analyzing the syndrome of RGC-rings gained in the first step of HYP-DIAG and analyze the proposed algorithm.

• Journal of Pharmaceutical Investigation
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• v.26 no.1
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• pp.55-59
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• 1996

The crystal structure of diflunisal, 2',4'-difluoro-4-hydroxy-3-biphenyl-carboxylic acid, was determined by single crystal X-ray diffraction technique. The compound was recrystallized from a mixture of acetone and water in monoclinic, space group C2/c, with $a\;=\;34.666(6),\;b\;=\;3.743(1),\;c\;=\;20.737(4)\;{\AA},\;{\beta}=\;110.57(2)^{\circ}$, and Z = 8. The calculated density is $1.324\;g/cm^3$. The structure was solved by the direct method and refined by full matrix least-squares procedure to the final R value of 0.045 for 1299 observed reflections. It was found that the molecules in the crystal are partially disordered, that is, the two equivalent conformers $(180^{\circ}$ rotated ones through C(1)-C(7)) are packed alternatively without regular symmetry or sequence. The two phenyl rings of the biphenyl group is tilted to each other by the dihedral angle of $43.3^{\circ}$. The carboxyl group at the salicylic moiety is just coplanar to the phenyl ring, and the planarity of this salicylic moiety is stabilized by an intramolecular hydrogen bond of O(3)-H(O3) O(2). The molecules are dimerized through the intermolecular hydrogen bonds at the carboxyl group in the crystal.

• Journal for History of Mathematics
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• v.19 no.2
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• pp.125-140
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• 2006

In 1933, I. Schur introduced a Schur ring in connection with permutation group and regular subgroup. After then, it was studied mostly for purely group theoretical purposes. In 1970s, Klin and Poschel initiated its usage in the investigation of graphs, especially for Cayley and circulant graphs. Nowadays it is known that Schur ring is one of the best way to enumerate Cayley graphs. In this paper we study the origin of Schur ring back to 1933 and keep trace its evolution to graph theory and combinatorics.

• Bulletin of the Korean Mathematical Society
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• v.20 no.1
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• pp.45-49
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• 1983

If R is ring and M is a right (or left) R-module, then M is called a faithful R-module if, for some a in R, x.a=0 for all x.mem.M then a=0. In [4], R.E. Johnson defines that M is a prime module if every non-zero submodule of M is faithful. Let us define that M is of prime type provided that M is faithful if and only if every non-zero submodule is faithful. We call a right (left) ideal I of R is of prime type if R/I is of prime type as a R-module. This is equivalent to the condition that if xRy.subeq.I then either x.mem.I ro y.mem.I (see [5:3:1]). It is easy to see that in case R is a commutative ring then a right or left ideal of a prime type is just a prime ideal. We have defined in [5], that a chain of right ideals of prime type in a ring R is a finite strictly increasing sequence I$_{0}$.contnd.I$_{1}$.contnd....contnd.I$_{n}$; the length of the chain is n. By the right dimension of a ring R, which is denoted by dim, R, we mean the supremum of the length of all chains of right ideals of prime type in R. It is an integer .geq.0 or .inf.. The left dimension of R, which is denoted by dim$_{l}$ R is similarly defined. It was shown in [5], that dim$_{r}$R=0 if and only if dim$_{l}$ R=0 if and only if R modulo the prime radical is a strongly regular ring. By "a strongly regular ring", we mean that for every a in R there is x in R such that axa=a=a$^{2}$x. It was also shown that R is a simple ring if and only if every right ideal is of prime type if and only if every left ideal is of prime type. In case, R is a (right or left) primitive ring then dim$_{r}$R=n if and only if dim$_{l}$ R=n if and only if R.iden.D$_{n+1}$ , n+1 by n+1 matrix ring on a division ring D. in this paper, we establish the following results: (1) If R is prime ring and dim$_{r}$R=n then either R is a righe Ore domain such that every non-zero right ideal of a prime type contains a non-zero minimal prime ideal or the classical ring of ritght quotients is isomorphic to m*m matrix ring over a division ring where m.leq.n+1. (b) If R is prime ring and dim$_{r}$R=n then dim$_{l}$ R=n if dim$_{l}$ R=n if dim$_{l}$ R<.inf. (c) Let R be a principal right and left ideal domain. If dim$_{r}$R=1 then R is an unique factorization domain.TEX>R=1 then R is an unique factorization domain.