• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.51-69
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• 2017

We, in this paper, study the commutativity of skew polynomials at zero as a generalization of an ${\alpha}-rigid$ ring, introducing the concept of strongly skew reversibility. A ring R is be said to be strongly ${\alpha}-skew$ reversible if the skew polynomial ring $R[x;{\alpha}]$ is reversible. We examine some characterizations and extensions of strongly ${\alpha}-skew$ reversible rings in relation with several ring theoretic properties which have roles in ring theory.

• Communications of the Korean Mathematical Society
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• v.26 no.3
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• pp.339-348
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• 2011

A ring R is called symmetric, if abc = 0 implies acb = 0 for a, b, c ${\in}$ R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R=I (resp. R/$\sqrt{I}$) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then $T_n$(R) and R[x]=($x^n$) are radically-symmetric, where ($x^n$) is the ideal of R[x] generated by $x^n$. Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) (${\alpha}$, ${\delta}$)-compatible ring, then R[x; ${\alpha}$, ${\delta}$] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].

• Korean Journal of Mathematics
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• v.18 no.2
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• pp.119-132
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• 2010

P. M. Cohn called a ring R reversible if whenever ab = 0, then ba = 0 for $a,b{\in}R$. In this paper, we study an extension of a reversible ring with its endomorphism. An endomorphism ${\alpha}$ of a ring R is called strong right (resp., left) reversible if whenever $a{\alpha}(b)=0$ (resp., ${\alpha}(a)b=0$) for $a,b{\in}R$, ba = 0. A ring R is called strong right (resp., left) ${\alpha}$-reversible if there exists a strong right (resp., left) reversible endomorphism ${\alpha}$ of R, and the ring R is called strong ${\alpha}$-reversible if R is both strong left and right ${\alpha}$-reversible. We investigate characterizations of strong ${\alpha}$-reversible rings and their related properties including extensions. In particular, we show that every semiprime and strong ${\alpha}$-reversible ring is ${\alpha}$-rigid and that for an ${\alpha}$-skew Armendariz ring R, the ring R is reversible and strong ${\alpha}$-reversible if and only if the skew polynomial ring $R[x;{\alpha}]$ of R is reversible.

• Journal of the Korean Chemical Society
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• v.37 no.3
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• pp.344-350
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• 1993

25,26,27,28-Tetraacetoxycalix[4]arene·monohydrate is orthorhombic, space group Pbca with a = 14.979(4), b = 15.154(4), c = 27.890(3) ${\AA}$, Z = 8, V = 6330.6 ${\AA}^{-3}$, D$_c$ = 1.28 $g{\cdot}cm^{-3}$, (Mo K${\alpha}$) = 0.71069 ${\AA}$, ${\mu}$ = 0.86 cm$^{-1}$, F(000) = 2600, and R = 0.069 for 3376 unique observed reflections with I > 1.0 ${\sigma}$(I). The structure was solved by direct methods and refined by cascade diagonal least-squares refinement. All the C-H bond lengths(= 0.96 ${\AA}$), the methyl groups and the methylene groups are fixed and refined as the rigid groups with ideal geometry. The macrocycle exists in the 1,3 alternate conformation (by Conforth) making the angles of 110.7, 684, 113.7 and 68.8$^{\circ}$ between the benzene rings and the methylenic mean plane, and four each acetoxy groups are twisted away from their own benzene rings with the angles of 68.2, 97.6, 78.9 and 71.3$^{\circ}$, respectively. The relative dihedral angles between two opposite side of the benzene rings are 135.6$^{\circ}$ for the rings (1) and (3) and 135.2$^{\circ}$ for (2) and (4). A water molecule which has nearly the same height of the methylenic plane of the macrocycle in the c-axis, is located within the distances of 2.942(5) ${\AA}$ from the O(8) atom of the carbonyl group and 2.901 ${\AA}$ from, another O(2)(1/2-x, -1/2＋y, z). The shortest contact between the molecule is 3.193 ${\AA}$ from the O(4) to the C(3)(1/2＋x, 1/2-y,-z).