• Title, Summary, Keyword: homogeneous ideal

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SOME FAMILIES OF IDEAL-HOMOGENEOUS POSETS

  • Chae, Gab-Byung;Cheong, Minseok;Kim, Sang-Mok
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.971-983
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    • 2016
  • A partially ordered set P is ideal-homogeneous provided that for any ideals I and J, if $$I{\sim_=}_{\sigma}J$$, then there exists an automorphism ${\sigma}^*$ such that ${\sigma}^*{\mid}_I={\sigma}$. Behrendt [1] characterizes the ideal-homogeneous partially ordered sets of height 1. In this paper, we characterize the ideal-homogeneous partially ordered sets of height 2 and nd some families of ideal-homogeneous partially ordered sets.

GRADED INTEGRAL DOMAINS IN WHICH EACH NONZERO HOMOGENEOUS IDEAL IS DIVISORIAL

  • Chang, Gyu Whan;Hamdi, Haleh;Sahandi, Parviz
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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    • pp.1041-1057
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    • 2019
  • Let ${\Gamma}$ be a nonzero commutative cancellative monoid (written additively), $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}$ $R_{\alpha}$ be a ${\Gamma}$-graded integral domain with $R_{\alpha}{\neq}\{0\}$ for all ${\alpha}{\in}{\Gamma}$, and $S(H)=\{f{\in}R{\mid}C(f)=R\}$. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if R is integrally closed, then R is a homogeneously divisorial domain if and only if $R_{S(H)}$ is an h-local $Pr{\ddot{u}}fer$ domain whose maximal ideals are invertible, if and only if R satisfies the following four conditions: (i) R is a graded-$Pr{\ddot{u}}fer$ domain, (ii) every homogeneous maximal ideal of R is invertible, (iii) each nonzero homogeneous prime ideal of R is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of R has only finitely many minimal prime ideals. We also show that if R is a graded-Noetherian domain, then R is a homogeneously divisorial domain if and only if $R_{S(H)}$ is a divisorial domain of (Krull) dimension one.

A Comparative Quantitative Analysis of IDEAL (Iterative Decomposition of Water and Fat with Echo Asymmetry and Least Squares Estimation) and CHESS (Chemical Shift Selection Suppression) Technique in 3.0T Musculoskeletal MRI

  • Kim, Myoung-Hoon;Cho, Jae-Hwan;Shin, Seong-Gyu;Dong, Kyung-Rae;Chung, Woon-Kwan;Park, Tae-Hyun;Ahn, Jae-Ouk;Park, Cheol-Soo;Jang, Hyon-Chol;Kim, Yoon-Shin
    • Journal of Magnetics
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    • v.17 no.2
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    • pp.145-152
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    • 2012
  • Patients who underwent hip arthroplasty using the conventional fat suppression technique (CHESS) and a new technique (IDEAL) were compared quantitatively to assess the effectiveness and usefulness of the IDEAL technique. In 20 patients who underwent hip arthroplasty from March 2009 to December 2010, fat suppression T2 and T1 weighted images were obtained on a 3.0T MR scanner using the CHESS and IDEAL techniques. The level of distortion in the area of interest, the level of the development of susceptibility artifacts, and homogeneous fat suppression were analyzed from the acquired images. Quantitative analysis revealed the IDEAL technique to produce a lower level of image distortion caused by the development of susceptibility artifacts due to metal on the acquired images compared to the CHESS technique. Qualitative analysis of the anterior area revealed the IDEAL technique to generate fewer susceptibility artifacts than the CHESS technique but with homogeneous fat suppression. In the middle area, the IDEAL technique generated fewer susceptibility artifacts than the CHESS technique but with homogeneous fat suppression. In the posterior area, the IDEAL technique generated fewer susceptibility artifacts than the CHESS technique. Fat suppression was not statistically different, and the two techniques achieved homogeneous fat suppression. In conclusion, the IDEAL technique generated fewer susceptibility artifacts caused by metals and less image distortion than the CHESS technique. In addition, homogeneous fat suppression was feasible. In conclusion, the IDEAL technique generates high quality images, and can provide good information for diagnosis.

GRADED INTEGRAL DOMAINS AND PRÜFER-LIKE DOMAINS

  • Chang, Gyu Whan
    • Journal of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.1733-1757
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    • 2017
  • Let $R={\oplus}_{{\alpha}{\in}{\Gamma}}R_{\alpha}$ be an integral domain graded by an arbitrary torsionless grading monoid ${\Gamma}$, ${\bar{R}}$ be the integral closure of R, H be the set of nonzero homogeneous elements of R, C(f) be the fractional ideal of R generated by the homogeneous components of $f{\in}R_H$, and $N(H)=\{f{\in}R{\mid}C(f)_v=R\}$. Let $R_H$ be a UFD. We say that a nonzero prime ideal Q of R is an upper to zero in R if $Q=fR_H{\cap}R$ for some $f{\in}R$ and that R is a graded UMT-domain if each upper to zero in R is a maximal t-ideal. In this paper, we study several ring-theoretic properties of graded UMT-domains. Among other things, we prove that if R has a unit of nonzero degree, then R is a graded UMT-domain if and only if every prime ideal of $R_{N(H)}$ is extended from a homogeneous ideal of R, if and only if ${\bar{R}}_{H{\backslash}Q}$ is a graded-$Pr{\ddot{u}}fer$ domain for all homogeneous maximal t-ideals Q of R, if and only if ${\bar{R}}_{N(H)}$ is a $Pr{\ddot{u}}fer$ domain, if and only if R is a UMT-domain.

A GORENSTEIN IDEAL OF CODIMENSION 4

  • Shin, Yong-Su
    • Journal of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.135-147
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    • 1997
  • Let k be an infinite field and let $X = {P_1, \cdots, P_s}$ be a set of s-distinct points in $P^n$. We denote by $I(X)$ the defining ideal of $X$ in the polynomial ring $R = k[x_0, \cdots, x_n]$ and by A the homogeneous coordinate ring of $X, A = \sum_{t = 0}^{\infty} A_t$.

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ON GRADED 2-ABSORBING PRIMARY AND GRADED WEAKLY 2-ABSORBING PRIMARY IDEALS

  • Al-Zoubi, Khaldoun;Sharafat, Nisreen
    • Journal of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.675-684
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    • 2017
  • Let G be a group with identity e and let R be a G-graded ring. In this paper, we introduce and study graded 2-absorbing primary and graded weakly 2-absorbing primary ideals of a graded ring which are different from 2-absorbing primary and weakly 2-absorbing primary ideals. We give some properties and characterizations of these ideals and their homogeneous components.

Improvement of Fat Suppression and Artifact Reduction Using IDEAL Technique in Head and Neck MRI at 3T

  • Hong, Jin Ho;Lee, Ha Young;Kang, Young Hye;Lim, Myung Kwan;Kim, Yeo Ju;Cho, Soon Gu;Kim, Mi Young
    • Investigative Magnetic Resonance Imaging
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    • v.20 no.1
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    • pp.44-52
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    • 2016
  • Purpose: To quantitatively and qualitatively compare fat-suppressed MRI quality using iterative decomposition of water and fat with echo asymmetry and least-squares estimation (IDEAL) with that using frequency selective fat-suppression (FSFS) T2- and postcontrast T1-weighted fast spin-echo images of the head and neck at 3T. Materials and Methods: The study was approved by our Institutional Review Board. Prospective MR image analysis was performed in 36 individuals at a single-center. Axial fat suppressed T2- and postcontrast T1-weighted images with IDEAL and FSFS were compared. Visual assessment was performed by two independent readers with respect to; 1) metallic artifacts around oral cavity, 2) susceptibility artifacts around upper airway, paranasal sinus, and head-neck junction, 3) homogeneity of fat suppression, 4) image sharpness, 5) tissue contrast of pathologies and lymph nodes. The signal-to-noise ratios (SNR) for each image sequence were assessed. Results: Both IDEAL fat suppressed T2- and T1-weighted images significantly reduced artifacts around airway, paranasal sinus, and head-neck junction, and significantly improved homogeneous fat suppression in compared to those using FSFS (P < 0.05 for all). IDEAL significantly decreased artifacts around oral cavity on T2-weighted images (P < 0.05, respectively) and improved sharpness, lesion-to-tissue, and lymph node-to-tissue contrast on T1-weighted images (P < 0.05 for all). The mean SNRs were significantly improved on both T1- and T2-weighted IDEAL images (P < 0.05 for all). Conclusion: IDEAL technique improves image quality in the head and neck by reducing artifacts with homogeneous fat suppression, while maintaining a high SNR.