• Bulletin of the Korean Mathematical Society
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• v.31 no.2
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• pp.163-165
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• 1994

Modules which satisfy the converse of Schur's lemma have been studied by many authors. In [6], R. Ware proved that a projective module P over a semiprime ring R is irreducible if and only if En $d_{R}$(P) is a division ring. Also, Y. Hirano and J.K. Park proved that a torsionless module M over a semiprime ring R is irreducible if and only if En $d_{R}$(M) is a division ring. In case R is a commutative ring, we obtain the following: An R-module M is irreducible if and only if En $d_{R}$(M) is a division ring and M is a multiplication R-module. Throughout this paper, R is commutative ring with identity and all modules are unital left R-modules. Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for each submodule N of M, there exists and ideal I of R such that N=IM. Cyclic R-modules are multiplication modules. In particular, irreducible R-modules are multiplication modules.dules.

• Journal of the Korean Mathematical Society
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• v.51 no.1
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• pp.87-98
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• 2014

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, say ${\Gamma}(M)$, such that when M = R, ${\Gamma}(M)$ is exactly the classic zero-divisor graph. Many well-known results by D. F. Anderson and P. S. Livingston, in [5], and by D. F. Anderson and S. B. Mulay, in [6], have been generalized for ${\Gamma}(M)$ in the present article. We show that ${\Gamma}(M)$ is connected with $diam({\Gamma}(M)){\leq}3$. We also show that for a reduced module M with $Z(M)^*{\neq}M{\backslash}\{0\}$, $gr({\Gamma}(M))={\infty}$ if and only if ${\Gamma}(M)$ is a star graph. Furthermore, we show that for a finitely generated semisimple R-module M such that its homogeneous components are simple, $x,y{\in}M{\backslash}\{0\}$ are adjacent if and only if $xR{\cap}yR=(0)$. Among other things, it is also observed that ${\Gamma}(M)={\emptyset}$ if and only if M is uniform, ann(M) is a radical ideal, and $Z(M)^*{\neq}M{\backslash}\{0\}$, if and only if ann(M) is prime and $Z(M)^*{\neq}M{\backslash}\{0\}$.

• Honam Mathematical Journal
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• v.27 no.1
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• pp.69-76
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• 2005

The aim of this work is to generalize $L_1$-approximation in order to apply them to a discrete approximation. In $L_1$-approximation, we use the norm given by $${\parallel}f{\parallel}_1={\int}{\mid}f{\mid}d{\mu}$$ where ${\mu}$ a non-atomic positive measure. In this paper, we go to the other extreme and consider measure ${\mu}$ which is purely atomic. In fact we shall assume that ${\mu}$ has exactly m atoms. For any ${\ell}$-tuple $b^1,\;{\cdots},\;b^{\ell}{\in}{\mathbb{R}}^m$, we defined the ${\ell}^m_1{w}$-norn, and consider $s^*{\in}S$ such that, for any $b^1,\;{\cdots},\;b^{\ell}{\in}{\mathbb{R}}^m$, $$\array{min&max\\{s{\in}S}&{1{\leq}i{\leq}{\ell}}}\;{\parallel}b^i-s{\parallel}_w$$, where S is a n-dimensional subspace of ${\mathbb{R}}^m$. The $s^*$ is called the Chebyshev center or a discrete simultaneous ${\ell}^m_1$-approximation from the finite dimensional subspace.

• Journal of Bio-Environment Control
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• v.14 no.4
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• pp.269-274
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• 2005

The purpose of this experiment was to investigate optimal ionic concentration of nutrient solution for water culture of young welsh onion (Allium fistulosum). For the purpose of clarification of optimal nutrient concentration to maximize growth of young welsh onion, different nutrient concentrations of Yamazaki's solution for welsh onion seedling $(NO_3^--N\;9.0,\;NH_4^+-N\;3.0,\;PO_4^{3-}-P\;6.0,\;K^+7.0,\;Ca^{2+}\;2.0,\;Mg^{2+}\;2.0,\;and\;SO_4^{2-}-S\;4.4me{\cdot}L^{-1})$ which selected by prior experiment were treated as 0.6, 1.2, 1.8, and $2.4dS{\cdot}m^{-1}$. Increments of fresh weight, dry weight and top length were the highest in 1.2 and, in the next, were placed by the order of 1.8, 2.4, and $0.6dS{\cdot}m^{-1}$ The regression coefficients for the maximal growth of fresh weight of cv. 'Geurnjanguedaepa' and 'Tokyokuro' were $y=-42.091x^2+171.79x+11.047 (R^2=0.8946,\; R=0.9458^*)\;and\;y=-50.069x2+157.58x+15.414(R^2=0.9343,\;R=0.9692^{**})$, respectively, and optimal EC levels according to regression coefficients were 1.68 and $1.57dS{\cdot}m^{-1}$. As the conclusions, optimal nutrient levels far young welsh onion were $1.2dS{\cdot}m^{-1}$ EC in the early growth stage and $1.6\~l .7dS{\cdot}m^{-1}$ in the later growth stage.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1221-1233
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• 2021

In this paper, we introduce and investigate a new class of modules that is closely related to the class of Noetherian modules. Let R be a commutative ring and M be an R-module. We say that M is an r-Noetherian module if every r-submodule of M is finitely generated. Also, we call the ring R to be an r-Noetherian ring if R is an r-Noetherian R-module, or equivalently, every r-ideal of R is finitely generated. We show that many properties of Noetherian modules are also true for r-Noetherian modules. Moreover, we extend the concept of weakly Noetherian rings to the category of modules and we characterize Noetherian modules in terms of r-Noetherian and weakly Noetherian modules. Finally, we use the idealization construction to give non-trivial examples of r-Noetherian rings that are not Noetherian.

• Elastomers and Composites
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• v.29 no.1
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• pp.18-29
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• 1994

The purpose of this study is to investigate reinforced effect between silica treated by coupling agents and rubber matrix under the configuration chemical bonds, and the effect of silica particles coated by organic polymers using aluminum chloride as the catalyst. In vulcanization characteristies were tested by Curastometer. The M-series vulcanizates were reached to the fastest optimum cure $time(t_{90})$ and R-series vulcanizates with the same formula had the shorted optimum cure times. Tensile characteristics measuring with a tensile tester revealed that the M-series vulcanizate was the best in the physical properties, such as tensile strength. In 100% modulus, however, the S-series vulcanizates appeared to be better than the other vulcanizates. Also, hardness showed the following order : S-series>R-series>M-series with the order of elongation R-series>M-series>S-series. In SEM test, shapes of chemical treated silicas were observed. The dispersion of filler in the SBR composite appeard uniformly. In RDS test for the dynamic characteristics, G' indicates that S-3 shows the highest value with the next order M-3>R-3, and the order of damping values are as followe: M-3>M-3>R-3.

• Journal of the Korean Mathematical Society
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• v.44 no.2
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• pp.373-391
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• 2007

We study an (n+3)($n\;{\geq}\;7-dimensional$ real submanifold of a (4m+3)-unit sphere $S^{4m+3}$ with Sasakian 3-structure induced from the canonical quaternionic $K\"{a}hler$ structure of quaternionic (m+1)-number space $Q^{m+1}$, and especially determine contact three CR-submanifolds with (p-1) contact three CR-dimension under the equality conditions given in (4.1), where p = 4m - n denotes the codimension of the submanifold. Also we provide necessary conditions concerning sectional curvature in order that a compact contact three CR-submanifold of (p-1) contact three CR-dimension in $S^{4m+3}$ is the model space $S^{4n_1+3}(r_1){\times}S^{4n_2+3}(r_2)$ for some portion $(n_1,\;n_2)$ of (n-3)/4 and some $r_1,\;r_2$ with $r^{2}_{1}+r^{2}_{2}=1$.

• Natural Product Sciences
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• v.10 no.3
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• pp.124-128
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• 2004

Previously, $(1R,9S)-{\beta}-Hydrastine$ hydrochloride has been found to lower dopamine content in PC12 cells (Kim et al., 20001). In this study, the effects of $(1R,9S)-{\beta}-Hydrastine$ hydrochloride on L-DOPA-induced cytotoxicity in PC12 cells were investigated. Treatment with $(1R,9S)-{\beta}-Hydrastine$ hydrochloride at concentrations higher than $500\;{\mu}M$ caused cytotoxicity in PC12 cells. In addition, $(1R,9S)-{\beta}-Hydrastine$ hydrochloride at non-cytotoxic or cytotoxic concentrations significantly enhanced L-DOPA-induced cytotoxicity (L-DOPA concentration, $50\;{\mu}M$). Treatment of PC12 cells with $750\;{\mu}M$ $-1R,9S)-{\beta}-Hydrastine$ hydrochloride and $50\;{\mu}M$ L-DOPA, alone or in combination, also induced cell death via a mechanism which exhibited morphological and biochemical characteristics of apoptosis, including chromatin condensation and membrane blebbing. Exposure of PC12 cells to $(1R,9S)-{\beta}-Hydrastine$ hydrochloride, L-DOPA and $(1R,9S)-{\beta}-Hydrastine$ hydrochloride plus L-DOPA for 48 h resulted in a marked increase in the cell loss and percentage of apoptotic cells compared with exposure for 24 h. These data indicate that $(1R,9S)-{\beta}-Hydrastine$hydrochloride at higher concentration ranges aggravates L-DOPA-induced neurotoxicity cytotoxicity in PC12 cells. Therefore, it is proposed that the long-term L-DOPA therapeutic patients with $(1R,9S)-{\beta}-Hydrastine$ hydrochloride could be checked for the adverse symptoms.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.885-902
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• 2016

In this paper we begin the study of $L_k$-2-type hypersurfaces of a hypersphere ${\mathbb{S}}^{n+1}{\subset}{\mathbb{R}}^{n+2}$ for $k{\geq}1$ Let ${\psi}:M^3{\rightarrow}{\mathbb{S}}^4$ be an orientable $H_k$-hypersurface, which is not an open portion of a hypersphere. Then $M^3$ is of $L_k$-2-type if and only if $M^3$ is a Clifford tori ${\mathbb{S}}^1(r_1){\times}{\mathbb{S}}^2(r_2)$, $r^2_1+r^2_2=1$, for appropriate radii, or a tube $T^r(V^2)$ of appropriate constant radius r around the Veronese embedding of the real projective plane ${\mathbb{R}}P^2({\sqrt{3}})$.

• Journal of the Korean Mathematical Society
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• v.52 no.6
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• pp.1323-1336
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• 2015

Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.