• Title/Summary/Keyword: group rings

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MATRIX RINGS AND ITS TOTAL RINGS OF FRACTIONS

  • Lee, Sang-Cheol
    • Honam Mathematical Journal
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    • v.31 no.4
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    • pp.515-527
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    • 2009
  • Let R be a commutative ring with identity. Then we prove $M_n(R)=GL_n(R)$ ${\cup}${$A{\in}M_n(R)\;{\mid}\;detA{\neq}0$ and det $A{\neq}U(R)$}${\cup}Z(M-n(R))$ where U(R) denotes the set of all units of R. In particular, it will be proved that the full matrix ring $M_n(F)$ over a field F is the disjoint union of the general linear group $GL_n(F)$ of degree n over the field F and the set $Z(M_n(F))$ of all zero-divisors of $M_n(F)$. Using the result and universal mapping property we prove that $M_n(F)$ is its total ring of fractions.

AN ALTERED GROUP RING CONSTRUCTION OF THE [24, 12, 8] AND [48, 24, 12] TYPE II LINEAR BLOCK CODE

  • Shefali Gupta;Dinesh Udar
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.829-844
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    • 2023
  • In this paper, we present a new construction for self-dual codes that uses the concept of double bordered construction, group rings, and reverse circulant matrices. Using groups of orders 2, 3, 4, and 5, and by applying the construction over the binary field and the ring F2 + uF2, we obtain extremal binary self-dual codes of various lengths: 12, 16, 20, 24, 32, 40, and 48. In particular, we show the significance of this new construction by constructing the unique Extended Binary Golay Code [24, 12, 8] and the unique Extended Quadratic Residue [48, 24, 12] Type II linear block code. Moreover, we strengthen the existing relationship between units and non-units with the self-dual codes presented in [10] by limiting the conditions given in the corollary. Additionally, we establish a relationship between idempotent and self-dual codes, which is done for the first time in the literature.

On Graded 2-Absorbing and Graded Weakly 2-Absorbing Primary Ideals

  • Soheilnia, Fatemeh;Darani, Ahmad Yousefian
    • Kyungpook Mathematical Journal
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    • v.57 no.4
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    • pp.559-580
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    • 2017
  • Let G be an arbitrary group with identity e and let R be a G-graded ring. In this paper, we define the concept of graded 2-absorbing and graded weakly 2-absorbing primary ideals of commutative G-graded rings with non-zero identity. A number of results and basic properties of graded 2-absorbing primary and graded weakly 2-absorbing primary ideals are given.

THE HILBERT-KUNZ MULTIPLICITY OF TWO-DIMENSIONAL TORIC RINGS

  • Choi, Sang-Ki;Hong, Seok-Young
    • Journal of the Korean Mathematical Society
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    • v.40 no.2
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    • pp.169-177
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    • 2003
  • Recently, K. Watanabe Showed that the Hilbert-Kunz multiplicity of a toric ring is a rational number. In this paper we give an explicit formula to compute the Hilbert-Kunz multiplicity of two-dimensional toric rings. This formula also shows that the Hilbert-Kunz multiplicity of a two-dimensional non-regular toric ring is at least 3/2.

Synthesis and X-ray Crystallographic Characterization of Spiro Orthocarbonates

  • Park Young Ja;No Kwang Hyun;Kim Ju Hee;Suh Il-Hwan
    • Bulletin of the Korean Chemical Society
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    • v.13 no.4
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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) ${\AA}$.

Hexaphenylbenzene $C_6(C_6H_5)_6$

  • Kim Young-Sang;Ko Jaejung;Kang Sang Ook;Han Won-Sik;Jeong Jae-Ho;Suh Il-Hwan
    • Korean Journal of Crystallography
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    • v.16 no.1
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    • pp.1-5
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    • 2005
  • The structure of the title compound has been determined by single-crystal X-ray diffraction work. The crystals are orthorhombic, space group $Pna2_1$ with a=11.095(3), b=21.834(7), c=12.574(4) $\AA$, and R1=0.0667. The average carbon bond length in aromaticity In the molecule is 1.386(1) $\AA$ and the average single bond length linking the central benEene ring and peripheral phenyl rings is 1.491(3) $\AA$. The average dihedral angle between the central benzene ring and each of six peripheral phenyl rings is $67.1(1)^{\circ}$ and the average dihedral angle between neighboring two phenyl rings is $55.0(1)^{\circ}$. Thus the molecule adopts a quasi-propeller configuration with approximate six-fold rotation symmetry.