• Kyungpook Mathematical Journal
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• 제51권2호
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• pp.177-185
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• 2011

A right ideal I of a ring R is small in case for every proper right ideal K of R, K + I ${\neq}$ = R. A right R-module M is called PS-injective if every R-homomorphism f : aR ${\rightarrow}$ M for every principally small right ideal aR can be extended to R ${\rightarrow}$ M. A ring R is called right PS-injective if R is PS-injective as a right R-module. We develop, in this article, PS-injectivity as a generalization of P-injectivity and small injectivity. Many characterizations of right PS-injective rings are studied. In light of these facts, we get several new properties of a right GPF ring and a semiprimitive ring in terms of right PS-injectivity. Related examples are given as well.

• 대한수학회논문집
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• 제19권2호
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• pp.211-217
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• 2004

In a ring $R_n(K,\;J)$ where K is a commutative ring with identity and J is an ideal of K, all prime ideals of $R_n(K,\;J)$ are of the form either $M_n(P)\;o;R_n(P,\;P\;{\cap}\;J)$. Therefore there is a one to one correspondence between prime ideals of K not containing J and prime ideals of $R_n(K,\;J)$.

• 대한수학회지
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• 제53권2호
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• pp.381-401
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• 2016

According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and ${\sigma}:M{\rightarrow}End(R)$ a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and ${\sigma}:M{\rightarrow}End(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).

• Kyungpook Mathematical Journal
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• 제46권4호
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• pp.503-507
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• 2006

Let X be a Tychonoff space and ${\sum}(X)$ the set of all the subrings of C(X) that contain $C^*(X)$. For any A(X) in ${\sum}(X)$ suppose $_{{\upsilon}A}X$ is the largest subspace of ${\beta}X$ containing X to which each function in A(X) can be extended continuously. Let us write A(X) ~ B(X) if and only if $_{{\upsilon}A}X=_{{\upsilon}B}X$, thereby defining an equivalence relation on ${\sum}(X)$. We have shown that an A(X) in ${\sum}(X)$ is isomorphic to C(Y ) for some space Y if and only if A(X) is the largest member of its equivalence class if and only if there exists a subspace T of ${\beta}X$ with the property that A(X)={$f{\in}C(X):f^*(p)$ is real for each $p$ in T}, $f^*$ being the unique continuous extension of $f$ in C(X) from ${\beta}X$ to $\mathbb{R}^*$, the one point compactification of $\mathbb{R}$. As a consequence it follows that if X is a realcompact space in which every $C^*$-embedded subset is closed, then C(X) is never isomorphic to any A(X) in ${\sum}(X)$ without being equal to it.

• Bulletin of the Korean Chemical Society
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• 제13권4호
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• pp.375-381
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• 1992

In this study we have synthesized two spiro orthocarbonates, which can be polymerized with volume expansion, and determined their crystal structures. The crystal data are as follows; 3,4,10,11-Di(9,10-dihydro-9,10-ethanoanthracenyl)- 1,6,8,13-tetraoxa-6.6-tridecane 5: a = 16.898 (1), b = 9.299 (1), c = 24.359 (2) ${\AA}$, $\beta$ = 123.73 $(7)^{\circ}$, space group P21/c and R = 0.073 for 2954 reflections; compound 8: a = 15.244 (4), b = 15.293 (3), c = 10.772 (3) $\AA$, ${\beta}$ = 99.45 $(2)^{\circ}$, space group P21/c and R = 0.082 for 2346 reflections. The seven-membered rings of compound 5 are chair forms and all the six-membered rings are boat shaped. For a six-membered spiro orthocarbonate, 3,9-Di(9-fluorenylidenyl)-1,4,6,9-tetraoxa-5,5-und ecane 8, fluorene groups [C(1) atom through C(13) atom] are planar within ${\pm}0.09{\AA}$ and the six-membered rings have chair conformations. The whole molecule has pseudo-C2 symmetry. The water molecules in the crystal are linked with each other through the hydrogen bond with distance of 2.790 (20) ${\AA}$.

• 대한본초학회지
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• 제28권4호
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• pp.63-69
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• 2013

Objectives : The purpose of present study was to investigate the vasorelaxant activities and mechanisms of action of the ethanol extract of P. yedoensis leaf (PYL) on isolated rat aortic rings. Methods : Dried P. yedoensis leaves were extracted 3 times with 100% ethanol for 3 h in a reflux apparatus. Isolated rat aortic rings were suspended in organ chambers containing 10 ml Krebs-Henseleit (K-H) solution. The rings were maintained at $37^{\circ}C$ and aerated with a mixture of 95% $O_2$ and 5% $CO_2$. Changes in their tension were recorded via isometric transducers connected to a data acquisition system. Results : PYL relaxed the contraction of aortic rings induced by phenylephrine (PE, 1 ${\mu}M$) or KCl (60 mM) in a concentration dependent manner. However, the vasorelaxant effects of PYL on endothelium-denuded aortic rings were lower than endothelium-intact aortic rings. And the vasorelaxant effects of PYL on endothelium-intact aortic rings were reduced by pre-treatment with $N{\omega}$-Nitro-L-arginine methyl ester (10 ${\mu}M$), methylene blue (10 ${\mu}M$), 1-H-[1,2,4]-oxadiazolo-[4,3-${\alpha}$]-quinoxalin-1-one (10 ${\mu}M$), tetraethylammonium (5 mM). In addition, PYL inhibited the contraction induced by extracellular $Ca^{2+}$ in endothelium-denuded aortic rings pre-contracted by PE or KCl in $Ca^{2+}$-free K-H solution. Conclusions : These results suggest that PYL exerts its vasorelaxant effects via the activation of Nitric Oxide (NO) formation by means of L-arginine and NO-cGMP pathways and via the blockage of receptor operated calcium channels, voltage dependent calcium channels and calcium-activated potassium channels.

• Korean Journal of Mathematics
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• 제22권1호
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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 $P{\upsilon}MD$.

• 대한수학회지
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• 제51권5호
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• pp.971-985
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• 2014

All modules considered in this note are over associative commutative rings with an identity element. We show that a ${\omega}$-local module M is Rad-supplemented if and only if M/P(M) is a local module, where P(M) is the sum of all radical submodules of M. We prove that ${\omega}$-local nonsmall submodules of a cyclic Rad-supplemented module are again Rad-supplemented. It is shown that commutative Noetherian rings over which every w-local Rad-supplemented module is supplemented are Artinian. We also prove that if a finitely generated Rad-supplemented module is cyclic or multiplication, then it is amply Rad-supplemented. We conclude the paper with a characterization of finitely generated amply Rad-supplemented left modules over any ring (not necessarily commutative).

• 대한수학회보
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• 제58권6호
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• pp.1387-1400
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• 2021

Suppose that $T=\(\array{A&0\\U&B}\)$ is a formal triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. Let ℭ₁ and ℭ₂ be two classes of left A-modules, 𝔇₁ and 𝔇₂ be two classes of left B-modules. We prove that (ℭ₁, ℭ₂) and (𝔇₁, 𝔇₂) are admissible balanced pairs if and only if (p(ℭ₁, 𝔇₁), h(ℭ₂, 𝔇₂) is an admissible balanced pair in T-Mod. Furthermore, we describe when ($P^{C_1}_{D_1}$, $I^{C_2}_{D_2}$) is an admissible balanced pair in T-Mod. As a consequence, we characterize when T is a left virtually Gorenstein ring.

• Kyungpook Mathematical Journal
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• 제60권2호
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• pp.415-421
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• 2020

In the proof of Theorem 3 on p.253 in [4], both right and left distributivity are assumed simultaneously which makes the proof invalid. We give a corrected proof for this theorem by introducing an extension of Lemma 2.2 in [2].